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x^y - y^x

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  • Décio Luiz Gazzoni Filho
    Paul Leyland maintains a table of primes of the form x^y + y^x. Has anyone looked into the form x^y - y^x, and if not, is there any reason or has it just been
    Message 1 of 29 , Apr 24, 2005
      Paul Leyland maintains a table of primes of the form x^y + y^x. Has anyone
      looked into the form x^y - y^x, and if not, is there any reason or has it
      just been overlooked?

      I gave a bit of thought to them, and other than the condition gcd(x,y) = 1,
      which was already known for x^y + y^x, I only came up with the following: x^y
      - y^x = -(y^x - x^y), and in particular x^y - y^x is positive if x < y (the
      sole exceptions are the trivial x = 1 ones and 2^3 - 3^2 = -1), so those two
      conditions cut the search space by 6/pi^2 = 0.61 and 0.5 respectively.

      I have exhaustively searched the range 1 < x < y <= 500 using a GP script with
      ispseudoprime() calls, and later proving the primality of PRPs using PRIMO.
      I'm currently using PFGW with the -f switch to search the range 1 < x,500 < y
      <= 2000. I can't search far beyond this without a proper sieve. Perhaps Mark
      Rodenkirch could patch MultiSieve to sieve this form? Since it already
      supports x^y + y^x I guess the changes involved would be trivial.

      Here are the proven primes:

      3^4 - 4^3
      2^5 - 5^2
      2^7 - 7^2
      6^7 - 7^6
      2^9 - 9^2
      3^10 - 10^3
      9^10 - 10^9
      8^11 - 11^8
      6^13 - 13^6
      12^13 - 13^12
      5^14 - 14^5
      11^14 - 14^11
      2^17 - 17^2
      2^19 - 19^2
      7^20 - 20^7
      7^24 - 24^7
      12^25 - 25^12
      5^26 - 26^5
      21^32 - 32^21
      6^35 - 35^6
      13^38 - 38^13
      31^42 - 42^31
      32^43 - 43^32
      44^45 - 45^44
      24^49 - 49^24
      9^50 - 50^9
      2^51 - 51^2
      32^51 - 51^32
      3^52 - 52^3
      2^53 - 53^2
      6^53 - 53^6
      23^60 - 60^23
      41^60 - 60^41
      15^68 - 68^15
      34^69 - 69^34
      24^71 - 71^24
      19^72 - 72^19
      11^80 - 80^11
      2^81 - 81^2
      2^83 - 83^2
      42^85 - 85^42
      50^87 - 87^50
      63^88 - 88^63
      81^88 - 88^81
      8^89 - 89^8
      14^89 - 89^14
      30^91 - 91^30
      58^93 - 93^58
      68^95 - 95^68
      20^97 - 97^20
      29^98 - 98^29
      81^98 - 98^81
      14^101 - 101^14
      74^109 - 109^74
      47^110 - 110^47
      3^112 - 112^3
      15^112 - 112^15
      6^115 - 115^6
      2^119 - 119^2
      63^124 - 124^63
      109^134 - 134^109
      98^135 - 135^98
      50^143 - 143^50
      6^145 - 145^6
      109^150 - 150^109
      115^168 - 168^115
      125^168 - 168^125
      54^169 - 169^54
      98^169 - 169^98
      20^173 - 173^20
      130^177 - 177^130
      150^179 - 179^150
      61^180 - 180^61
      30^181 - 181^30
      132^181 - 181^132
      103^182 - 182^103
      2^189 - 189^2
      95^198 - 198^95
      21^200 - 200^21
      8^201 - 201^8
      115^206 - 206^115
      24^211 - 211^24
      3^212 - 212^3
      11^212 - 212^11
      2^219 - 219^2
      63^220 - 220^63
      18^221 - 221^18
      145^222 - 222^145
      15^224 - 224^15
      2^227 - 227^2
      11^230 - 230^11
      109^234 - 234^109
      139^240 - 240^139
      66^245 - 245^66
      44^247 - 247^44
      57^250 - 250^57
      234^251 - 251^234
      255^268 - 268^255
      203^270 - 270^203
      219^272 - 272^219
      10^273 - 273^10
      93^278 - 278^93
      119^282 - 282^119
      18^283 - 283^18
      257^284 - 284^257
      255^292 - 292^255
      205^294 - 294^205
      56^295 - 295^56
      192^295 - 295^192
      2^301 - 301^2
      6^307 - 307^6
      117^310 - 310^117
      171^310 - 310^171
      12^325 - 325^12
      32^327 - 327^32
      95^332 - 332^95
      177^332 - 332^177
      321^334 - 334^321
      44^335 - 335^44
      137^338 - 338^137
      127^342 - 342^127
      3^346 - 346^3
      193^360 - 360^193
      96^365 - 365^96
      62^373 - 373^62
      240^383 - 383^240
      354^391 - 391^354
      128^397 - 397^128
      10^399 - 399^10
      342^403 - 403^342
      3^406 - 406^3
      71^408 - 408^71
      354^415 - 415^354
      104^417 - 417^104
      195^418 - 418^195
      337^420 - 420^337
      133^422 - 422^133
      42^425 - 425^42
      24^427 - 427^24

      Here are the pseudoprimes:

      174^439 - 439^174
      42^445 - 445^42
      2^455 - 455^2
      2^461 - 461^2
      101^462 - 462^101
      172^471 - 471^172
      24^481 - 481^24
      83^486 - 486^83
      384^487 - 487^384
      121^488 - 488^121
      173^488 - 488^173
      114^491 - 491^114
      329^494 - 494^329
      240^497 - 497^240
      455^498 - 498^455
      50^501 - 501^50
      3^512 - 512^3
      294^515 - 515^294
      485^516 - 516^485
      299^518 - 518^299
      390^527 - 527^390
      363^538 - 538^363
      398^539 - 539^398
      182^547 - 547^182
      475^552 - 552^475
      152^559 - 559^152
      360^559 - 559^360
      398^563 - 563^398
      14^579 - 579^14
      244^579 - 579^244
      323^584 - 584^323
      272^587 - 587^272
      332^589 - 589^332
      232^597 - 597^232
      323^606 - 606^323
      392^607 - 607^392
      215^608 - 608^215
      497^608 - 608^497
      2^623 - 623^2
      30^631 - 631^30
      532^633 - 633^532
      535^636 - 636^535
      302^641 - 641^302
      362^653 - 653^362
      14^655 - 655^14
      48^661 - 661^48
      119^668 - 668^119
      305^668 - 668^305
      629^668 - 668^629
      248^669 - 669^248
      116^675 - 675^116
      133^680 - 680^133
      301^692 - 692^301
      201^694 - 694^201
      171^704 - 704^171
      62^707 - 707^62
      446^715 - 715^446
      584^715 - 715^584
      49^720 - 720^49
      38^721 - 721^38
      240^721 - 721^240
      389^722 - 722^389
      81^724 - 724^81
      181^732 - 732^181
      38^735 - 735^38
      233^740 - 740^233
      398^741 - 741^398
      722^745 - 745^722
      649^774 - 774^649
      685^774 - 774^685
      691^774 - 774^691
      482^775 - 775^482
      113^782 - 782^113
      37^786 - 786^37
      334^789 - 789^334
      159^794 - 794^159
      325^798 - 798^325
      428^809 - 809^428
      21^814 - 814^21
      18^821 - 821^18
      282^821 - 821^282
      139^824 - 824^139
      12^833 - 833^12
      747^842 - 842^747
      338^859 - 859^338
      509^864 - 864^509
      181^878 - 878^181
      594^881 - 881^594
      883^884 - 884^883
      165^892 - 892^165
      472^897 - 897^472
      40^903 - 903^40
      272^903 - 903^272
      49^908 - 908^49
      398^913 - 913^398
      272^917 - 917^272
      866^925 - 925^866
      765^926 - 926^765
      847^930 - 930^847
      91^932 - 932^91
      740^933 - 933^740
      686^937 - 937^686
      702^941 - 941^702
      192^943 - 943^192
      770^943 - 943^770
      349^944 - 944^349
      373^944 - 944^373
      451^948 - 948^451
      841^950 - 950^841
      908^961 - 961^908
      291^962 - 962^291
      688^975 - 975^688
      8^977 - 977^8
      637^984 - 984^637
      38^985 - 985^38
      42^1003 - 1003^42
      849^1018 - 1018^849
      789^1030 - 1030^789
      408^1031 - 1031^408

      As anyone can check against Paul Leyland's tables at
      http://www.leyland.vispa.com/numth/primes/xyyx.htm, the form x^y - y^x is
      much more productive than x^y + y^x. For x,y < 1000, there are 254 x^y - y^x
      primes and only 87 x^y + y^x primes. Frankly, this a mistery to a me. I see
      no reason for them to be any different, even taking into account the fact
      that x^y - y^x < x^y + y^x: the ratio x^y/y^x is always huge, even in the
      most favorable case, x + 1 = y (here the ratio seems to be well approximated
      by (x+y)/2e ~ 0.184(x+y)), so that the probability that each form is prime,
      log(x^y + y^x) and log(x^y - y^x) are essentially the same.

      Interestingly, 2^9 + 9^2 and 2^9 - 9^2 is a `twin' pair, and the only one I've
      found yet. I'm not hoping to find a larger pair, but it would be a very
      welcome surprise.

      Décio
    • Kevin Acres
      Hi, I searched for a while using that form, but then switched to x^y +/- y^x +/- x*y. This gave fairly good results as regards the number of large PrP s found
      Message 2 of 29 , Apr 24, 2005
        Hi,

        I searched for a while using that form, but then switched to x^y +/- y^x
        +/- x*y. This gave fairly good results as regards the number of large
        PrP's found with PFGW.

        Kevin.

        At 06:16 AM 25/04/2005, Décio Luiz Gazzoni Filho wrote:

        >Paul Leyland maintains a table of primes of the form x^y + y^x. Has anyone
        >looked into the form x^y - y^x, and if not, is there any reason or has it
        >just been overlooked?
        >
        >I gave a bit of thought to them, and other than the condition gcd(x,y) = 1,
        >which was already known for x^y + y^x, I only came up with the following: x^y
        >- y^x = -(y^x - x^y), and in particular x^y - y^x is positive if x < y (the
        >sole exceptions are the trivial x = 1 ones and 2^3 - 3^2 = -1), so those two
        >conditions cut the search space by 6/pi^2 = 0.61 and 0.5 respectively.
        >
        >I have exhaustively searched the range 1 < x < y <= 500 using a GP script
        >with
        >ispseudoprime() calls, and later proving the primality of PRPs using PRIMO.
        >I'm currently using PFGW with the -f switch to search the range 1 < x,500 < y
        ><= 2000. I can't search far beyond this without a proper sieve. Perhaps Mark
        >Rodenkirch could patch MultiSieve to sieve this form? Since it already
        >supports x^y + y^x I guess the changes involved would be trivial.
        >
        >Here are the proven primes:
        >
        >3^4 - 4^3
        >2^5 - 5^2
        >2^7 - 7^2
        >6^7 - 7^6
        >2^9 - 9^2
        >3^10 - 10^3
        >9^10 - 10^9
        >8^11 - 11^8
        >6^13 - 13^6
        >12^13 - 13^12
        >5^14 - 14^5
        >11^14 - 14^11
        >2^17 - 17^2
        >2^19 - 19^2
        >7^20 - 20^7
        >7^24 - 24^7
        >12^25 - 25^12
        >5^26 - 26^5
        >21^32 - 32^21
        >6^35 - 35^6
        >13^38 - 38^13
        >31^42 - 42^31
        >32^43 - 43^32
        >44^45 - 45^44
        >24^49 - 49^24
        >9^50 - 50^9
        >2^51 - 51^2
        >32^51 - 51^32
        >3^52 - 52^3
        >2^53 - 53^2
        >6^53 - 53^6
        >23^60 - 60^23
        >41^60 - 60^41
        >15^68 - 68^15
        >34^69 - 69^34
        >24^71 - 71^24
        >19^72 - 72^19
        >11^80 - 80^11
        >2^81 - 81^2
        >2^83 - 83^2
        >42^85 - 85^42
        >50^87 - 87^50
        >63^88 - 88^63
        >81^88 - 88^81
        >8^89 - 89^8
        >14^89 - 89^14
        >30^91 - 91^30
        >58^93 - 93^58
        >68^95 - 95^68
        >20^97 - 97^20
        >29^98 - 98^29
        >81^98 - 98^81
        >14^101 - 101^14
        >74^109 - 109^74
        >47^110 - 110^47
        >3^112 - 112^3
        >15^112 - 112^15
        >6^115 - 115^6
        >2^119 - 119^2
        >63^124 - 124^63
        >109^134 - 134^109
        >98^135 - 135^98
        >50^143 - 143^50
        >6^145 - 145^6
        >109^150 - 150^109
        >115^168 - 168^115
        >125^168 - 168^125
        >54^169 - 169^54
        >98^169 - 169^98
        >20^173 - 173^20
        >130^177 - 177^130
        >150^179 - 179^150
        >61^180 - 180^61
        >30^181 - 181^30
        >132^181 - 181^132
        >103^182 - 182^103
        >2^189 - 189^2
        >95^198 - 198^95
        >21^200 - 200^21
        >8^201 - 201^8
        >115^206 - 206^115
        >24^211 - 211^24
        >3^212 - 212^3
        >11^212 - 212^11
        >2^219 - 219^2
        >63^220 - 220^63
        >18^221 - 221^18
        >145^222 - 222^145
        >15^224 - 224^15
        >2^227 - 227^2
        >11^230 - 230^11
        >109^234 - 234^109
        >139^240 - 240^139
        >66^245 - 245^66
        >44^247 - 247^44
        >57^250 - 250^57
        >234^251 - 251^234
        >255^268 - 268^255
        >203^270 - 270^203
        >219^272 - 272^219
        >10^273 - 273^10
        >93^278 - 278^93
        >119^282 - 282^119
        >18^283 - 283^18
        >257^284 - 284^257
        >255^292 - 292^255
        >205^294 - 294^205
        >56^295 - 295^56
        >192^295 - 295^192
        >2^301 - 301^2
        >6^307 - 307^6
        >117^310 - 310^117
        >171^310 - 310^171
        >12^325 - 325^12
        >32^327 - 327^32
        >95^332 - 332^95
        >177^332 - 332^177
        >321^334 - 334^321
        >44^335 - 335^44
        >137^338 - 338^137
        >127^342 - 342^127
        >3^346 - 346^3
        >193^360 - 360^193
        >96^365 - 365^96
        >62^373 - 373^62
        >240^383 - 383^240
        >354^391 - 391^354
        >128^397 - 397^128
        >10^399 - 399^10
        >342^403 - 403^342
        >3^406 - 406^3
        >71^408 - 408^71
        >354^415 - 415^354
        >104^417 - 417^104
        >195^418 - 418^195
        >337^420 - 420^337
        >133^422 - 422^133
        >42^425 - 425^42
        >24^427 - 427^24
        >
        >Here are the pseudoprimes:
        >
        >174^439 - 439^174
        >42^445 - 445^42
        >2^455 - 455^2
        >2^461 - 461^2
        >101^462 - 462^101
        >172^471 - 471^172
        >24^481 - 481^24
        >83^486 - 486^83
        >384^487 - 487^384
        >121^488 - 488^121
        >173^488 - 488^173
        >114^491 - 491^114
        >329^494 - 494^329
        >240^497 - 497^240
        >455^498 - 498^455
        >50^501 - 501^50
        >3^512 - 512^3
        >294^515 - 515^294
        >485^516 - 516^485
        >299^518 - 518^299
        >390^527 - 527^390
        >363^538 - 538^363
        >398^539 - 539^398
        >182^547 - 547^182
        >475^552 - 552^475
        >152^559 - 559^152
        >360^559 - 559^360
        >398^563 - 563^398
        >14^579 - 579^14
        >244^579 - 579^244
        >323^584 - 584^323
        >272^587 - 587^272
        >332^589 - 589^332
        >232^597 - 597^232
        >323^606 - 606^323
        >392^607 - 607^392
        >215^608 - 608^215
        >497^608 - 608^497
        >2^623 - 623^2
        >30^631 - 631^30
        >532^633 - 633^532
        >535^636 - 636^535
        >302^641 - 641^302
        >362^653 - 653^362
        >14^655 - 655^14
        >48^661 - 661^48
        >119^668 - 668^119
        >305^668 - 668^305
        >629^668 - 668^629
        >248^669 - 669^248
        >116^675 - 675^116
        >133^680 - 680^133
        >301^692 - 692^301
        >201^694 - 694^201
        >171^704 - 704^171
        >62^707 - 707^62
        >446^715 - 715^446
        >584^715 - 715^584
        >49^720 - 720^49
        >38^721 - 721^38
        >240^721 - 721^240
        >389^722 - 722^389
        >81^724 - 724^81
        >181^732 - 732^181
        >38^735 - 735^38
        >233^740 - 740^233
        >398^741 - 741^398
        >722^745 - 745^722
        >649^774 - 774^649
        >685^774 - 774^685
        >691^774 - 774^691
        >482^775 - 775^482
        >113^782 - 782^113
        >37^786 - 786^37
        >334^789 - 789^334
        >159^794 - 794^159
        >325^798 - 798^325
        >428^809 - 809^428
        >21^814 - 814^21
        >18^821 - 821^18
        >282^821 - 821^282
        >139^824 - 824^139
        >12^833 - 833^12
        >747^842 - 842^747
        >338^859 - 859^338
        >509^864 - 864^509
        >181^878 - 878^181
        >594^881 - 881^594
        >883^884 - 884^883
        >165^892 - 892^165
        >472^897 - 897^472
        >40^903 - 903^40
        >272^903 - 903^272
        >49^908 - 908^49
        >398^913 - 913^398
        >272^917 - 917^272
        >866^925 - 925^866
        >765^926 - 926^765
        >847^930 - 930^847
        >91^932 - 932^91
        >740^933 - 933^740
        >686^937 - 937^686
        >702^941 - 941^702
        >192^943 - 943^192
        >770^943 - 943^770
        >349^944 - 944^349
        >373^944 - 944^373
        >451^948 - 948^451
        >841^950 - 950^841
        >908^961 - 961^908
        >291^962 - 962^291
        >688^975 - 975^688
        >8^977 - 977^8
        >637^984 - 984^637
        >38^985 - 985^38
        >42^1003 - 1003^42
        >849^1018 - 1018^849
        >789^1030 - 1030^789
        >408^1031 - 1031^408
        >
        >As anyone can check against Paul Leyland's tables at
        >http://www.leyland.vispa.com/numth/primes/xyyx.htm, the form x^y - y^x is
        >much more productive than x^y + y^x. For x,y < 1000, there are 254 x^y - y^x
        >primes and only 87 x^y + y^x primes. Frankly, this a mistery to a me. I see
        >no reason for them to be any different, even taking into account the fact
        >that x^y - y^x < x^y + y^x: the ratio x^y/y^x is always huge, even in the
        >most favorable case, x + 1 = y (here the ratio seems to be well approximated
        >by (x+y)/2e ~ 0.184(x+y)), so that the probability that each form is prime,
        >log(x^y + y^x) and log(x^y - y^x) are essentially the same.
        >
        >Interestingly, 2^9 + 9^2 and 2^9 - 9^2 is a `twin' pair, and the only one
        >I've
        >found yet. I'm not hoping to find a larger pair, but it would be a very
        >welcome surprise.
        >
        >Décio
        >
        >
        >
        >Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
        >The Prime Pages : http://www.primepages.org/
        >
        >
        >Yahoo! Groups Links
        >
        >
        >
        >
      • Paul Leyland
        ... Hmm, interesting and something to which I ve not given any thought. One almost trivial observation is that differences of powers tend to be less likely to
        Message 3 of 29 , Apr 28, 2005
          On Sun, 2005-04-24 at 21:16, Décio Luiz Gazzoni Filho wrote:
          > Paul Leyland maintains a table of primes of the form x^y + y^x. Has anyone
          > looked into the form x^y - y^x, and if not, is there any reason or has it
          > just been overlooked?
          ...
          > As anyone can check against Paul Leyland's tables at
          > http://www.leyland.vispa.com/numth/primes/xyyx.htm, the form x^y - y^x is
          > much more productive than x^y + y^x. For x,y < 1000, there are 254 x^y - y^x
          > primes and only 87 x^y + y^x primes. Frankly, this a mistery to a me. I see
          > no reason for them to be any different, even taking into account the fact
          > that x^y - y^x < x^y + y^x: the ratio x^y/y^x is always huge, even in the
          > most favorable case, x + 1 = y (here the ratio seems to be well approximated
          > by (x+y)/2e ~ 0.184(x+y)), so that the probability that each form is prime,
          > log(x^y + y^x) and log(x^y - y^x) are essentially the same.

          Hmm, interesting and something to which I've not given any thought.

          One almost trivial observation is that differences of powers tend to be
          less likely to be prime than sums of powers because of the identity
          (a-b)(a+b) = a^2-b^2. However, the requirement that gcd(x,y) = 1
          prevents that identity being useful in the present circumstances. I
          don't immediately see why x^y-y^x, with the condition that gcd(x,y) = 1,
          should be more likely to be prime than x^y+y^x under the same
          restriction.

          I'll think about it. Thanks for drawing the problem my attention.


          Perhaps I should host another table of primes and strong pseudoptimes.
          Is there any support for this proposal


          Paul
        • Décio Luiz Gazzoni Filho
          ... However, consider the least prime of this form, 3^4 - 4^3 = 17. In this case, the expression could be rewritten as 9^2 - 8^2 = (9 - 8)(9 + 8) = 1*17 = 17.
          Message 4 of 29 , Apr 28, 2005
            On Thursday 28 April 2005 15:21, Paul Leyland wrote:
            > On Sun, 2005-04-24 at 21:16, Décio Luiz Gazzoni Filho wrote:
            > > Paul Leyland maintains a table of primes of the form x^y + y^x. Has
            > > anyone looked into the form x^y - y^x, and if not, is there any reason or
            > > has it just been overlooked?
            >
            > ...
            >
            > > As anyone can check against Paul Leyland's tables at
            > > http://www.leyland.vispa.com/numth/primes/xyyx.htm, the form x^y - y^x is
            > > much more productive than x^y + y^x. For x,y < 1000, there are 254 x^y -
            > > y^x primes and only 87 x^y + y^x primes. Frankly, this a mistery to a me.
            > > I see no reason for them to be any different, even taking into account
            > > the fact that x^y - y^x < x^y + y^x: the ratio x^y/y^x is always huge,
            > > even in the most favorable case, x + 1 = y (here the ratio seems to be
            > > well approximated by (x+y)/2e ~ 0.184(x+y)), so that the probability that
            > > each form is prime, log(x^y + y^x) and log(x^y - y^x) are essentially the
            > > same.
            >
            > Hmm, interesting and something to which I've not given any thought.
            >
            > One almost trivial observation is that differences of powers tend to be
            > less likely to be prime than sums of powers because of the identity
            > (a-b)(a+b) = a^2-b^2. However, the requirement that gcd(x,y) = 1
            > prevents that identity being useful in the present circumstances. I
            > don't immediately see why x^y-y^x, with the condition that gcd(x,y) = 1,
            > should be more likely to be prime than x^y+y^x under the same
            > restriction.

            However, consider the least prime of this form, 3^4 - 4^3 = 17. In this case,
            the expression could be rewritten as 9^2 - 8^2 = (9 - 8)(9 + 8) = 1*17 = 17.
            Checking through my logs unsurprisingly reveals that this is the only case
            where both x^y and y^x are squares.

            Décio


            [Non-text portions of this message have been removed]
          • Décio Luiz Gazzoni Filho
            ... I addressed that preemptively in the first email I sent. Have a look at the part emphasized above. Décio [Non-text portions of this message have been
            Message 5 of 29 , Apr 28, 2005
              On Thursday 28 April 2005 19:01, you wrote:
              > Décio Luiz Gazzoni Filho wrote:
              > >On Thursday 28 April 2005 15:21, Paul Leyland wrote:
              > >>On Sun, 2005-04-24 at 21:16, Décio Luiz Gazzoni Filho wrote:
              > >>>Paul Leyland maintains a table of primes of the form x^y + y^x. Has
              > >>>anyone looked into the form x^y - y^x, and if not, is there any reason
              > >>> or has it just been overlooked?
              > >>
              > >>...
              > >>
              > >>>As anyone can check against Paul Leyland's tables at
              > >>>http://www.leyland.vispa.com/numth/primes/xyyx.htm, the form x^y - y^x
              > >>> is much more productive than x^y + y^x. For x,y < 1000, there are 254
              > >>> x^y - y^x primes and only 87 x^y + y^x primes. Frankly, this a mistery

              ----- PAY ATTENTION TO THIS PART -----

              > >>> to a me. I see no reason for them to be any different, even taking into
              > >>> account the fact that x^y - y^x < x^y + y^x: the ratio x^y/y^x is
              > >>> always huge, even in the most favorable case, x + 1 = y (here the ratio
              > >>> seems to be well approximated by (x+y)/2e ~ 0.184(x+y)), so that the
              > >>> probability that each form is prime, log(x^y + y^x) and log(x^y - y^x)
              > >>> are essentially the same.

              ----- PAY ATTENTION TO THIS PART -----

              > >>
              > >>Hmm, interesting and something to which I've not given any thought.
              > >>
              > >>One almost trivial observation is that differences of powers tend to be
              > >>less likely to be prime than sums of powers because of the identity
              > >>(a-b)(a+b) = a^2-b^2. However, the requirement that gcd(x,y) = 1
              > >>prevents that identity being useful in the present circumstances. I
              > >>don't immediately see why x^y-y^x, with the condition that gcd(x,y) = 1,
              > >>should be more likely to be prime than x^y+y^x under the same
              > >>restriction.
              > >
              > >However, consider the least prime of this form, 3^4 - 4^3 = 17. In this
              > > case, the expression could be rewritten as 9^2 - 8^2 = (9 - 8)(9 + 8) =
              > > 1*17 = 17. Checking through my logs unsurprisingly reveals that this is
              > > the only case where both x^y and y^x are squares.
              > >
              > >Décio
              > >
              > >
              > >[Non-text portions of this message have been removed]
              >
              > Hi!
              > First it looks curious that x^y-y^x have more primes than x^y+y^x.
              > However, while the number in the plus-case grow fast for large and about
              > equal x and y,
              > in the minus-case they get smaller while x and y come more equal.
              > As the density of primes is larger for smaller number, this could be one
              > possible reason.

              I addressed that preemptively in the first email I sent. Have a look at the
              part emphasized above.

              Décio


              [Non-text portions of this message have been removed]
            • Mark Underwood
              ... y^x is ... x^y - y^x ... me. Pierre de Fermat solves de mystery. In the case of N = x^y + y^x (only), if x or y is one less than a prime p, then x^y + y^x
              Message 6 of 29 , Apr 28, 2005
                --- In primenumbers@yahoogroups.com, Décio Luiz Gazzoni Filho
                <decio@d...> wrote:
                >
                > As anyone can check against Paul Leyland's tables at
                > http://www.leyland.vispa.com/numth/primes/xyyx.htm, the form x^y -
                y^x is
                > much more productive than x^y + y^x. For x,y < 1000, there are 254
                x^y - y^x
                > primes and only 87 x^y + y^x primes. Frankly, this a mistery to a
                me.


                Pierre de Fermat solves de mystery.

                In the case of N = x^y + y^x (only), if x or y is one less than a
                prime p, then x^y + y^x contains a factor of p. (Assuming the other
                term x or y doesn't contain a factor of p) . Thus x^y + y^n cannot be
                prime if x or y is one less than a prime.

                Proof:

                Let x^y + y^x = N

                Let x = p-1

                Then y is odd. We assume that y has no factor of p.

                (p-1)^y + y^(p-1) = N

                -1 + y^(p-1) = N modp

                By Fermat,

                -1 + 1 = N modp

                0=N modp

                Thus N has a factor of p, so cannot be prime.


                Mark
              • Mark Underwood
                Just an additional tidbit. For the expression x^y - y^x to be prime, neither x nor y can be an even square. Proof: Assume x is an even square, so x = (2m)^2
                Message 7 of 29 , Apr 28, 2005
                  Just an additional tidbit. For the expression

                  x^y - y^x

                  to be prime, neither x nor y can be an even square.

                  Proof:

                  Assume x is an even square, so x = (2m)^2

                  Then

                  (2m)^2y - y^(2m*2m)

                  is clearly a difference of squares and is factored to

                  ((2m)^y - y^(2m))((2m)^y + y^(2m))

                  and so is not prime unless the first term is +/- 1. (And that might
                  only occur when m=1 and y=3.)

                  Mark
                • Décio Luiz Gazzoni Filho
                  ... Amazing work, Mark. I would never have thought of that. I m not sure that it can account for all of the differences, but regardless, it s a very
                  Message 8 of 29 , Apr 28, 2005
                    On Thursday 28 April 2005 21:25, you wrote:
                    > Pierre de Fermat solves de mystery.
                    >
                    > In the case of N = x^y + y^x (only), if x or y is one less than a
                    > prime p, then x^y + y^x contains a factor of p. (Assuming the other
                    > term x or y doesn't contain a factor of p) . Thus x^y + y^n cannot be
                    > prime if x or y is one less than a prime.
                    >
                    > Proof:
                    >
                    > Let x^y + y^x = N
                    >
                    > Let x = p-1
                    >
                    > Then y is odd. We assume that y has no factor of p.
                    >
                    > (p-1)^y + y^(p-1) = N
                    >
                    > -1 + y^(p-1) = N modp
                    >
                    > By Fermat,
                    >
                    > -1 + 1 = N modp
                    >
                    > 0=N modp
                    >
                    > Thus N has a factor of p, so cannot be prime.

                    Amazing work, Mark. I would never have thought of that. I'm not sure that it
                    can account for all of the differences, but regardless, it's a very
                    interesting discovery. I wonder if it can be generalized?

                    Of note, divisibility of x^y - y^x by p means that x^y == y^x mod p. Surely
                    this equation must have interesting properties?

                    Décio


                    [Non-text portions of this message have been removed]
                  • Mark Underwood
                    ... Yes I think there is more to it if one looks closely. Paul s x^y + y^x somehow loses about 66 percent of prime candidates compared to x^y - y^x. Since
                    Message 9 of 29 , Apr 28, 2005
                      --- In primenumbers@yahoogroups.com, Décio Luiz Gazzoni Filho
                      <decio@d...> wrote:
                      >I'm not sure that it
                      > can account for all of the differences, but regardless, it's a very
                      > interesting discovery. I wonder if it can be generalized?
                      >

                      Yes I think there is more to it if one looks closely. Paul's x^y + y^x
                      somehow loses about 66 percent of prime candidates compared to x^y -
                      y^x. Since there are 94 primes up to 500 and 250 even numbers, that
                      represents a loss of 'only' 39 percent by the p-1 aspect. There's
                      still a good percentage to account for. Perhaps as you say some kind o
                      of generalizing principle may account for much of the rest, I'm not
                      sure.

                      Mark
                    • Mark Underwood
                      ... p. Surely ... Sure. Of course, if x and y share a factor of p then x^y - y^x does. That s why we make sure x and y are relatively prime. But did we know
                      Message 10 of 29 , Apr 29, 2005
                        --- In primenumbers@yahoogroups.com, Décio Luiz Gazzoni Filho
                        <decio@d...> wrote:
                        > Of note, divisibility of x^y - y^x by p means that x^y == y^x mod
                        p. Surely
                        > this equation must have interesting properties?
                        >

                        Sure. Of course, if x and y share a factor of p then x^y - y^x does.
                        That's why we make sure x and y are relatively prime.

                        But did we know that x-1 and y-1 must also be relatively prime? It is
                        easy to show that if x-1 and y-1 share a factor of p, then x^y - y^x
                        does!

                        Futhermore, the even number -1 must also be relatively prime to the
                        other number + 1. It is easy to show that if the even number - 1
                        shares a factor p with the other (odd) number + 1, then x^y - y^n has
                        the same factor p as well!


                        In a related fashion with Paul's equation of x^y + y^x, not only must
                        x and y be relatively prime, but x+1 and y+1 must be relatively
                        prime. If x+1 and y+1 share a common factor of p, then so does x^y +
                        y^x !

                        Furthermore, the even number + 1 must be relatively prime to the
                        other number -1. If they share a common factor of p, then so does x^y
                        + y^x !


                        When we count up all the qualifying x,y pairs up to 500 for x^y - y^x
                        and for x^y + y^x (with Paul's equation, x and y's one less than a
                        prime are disqualified), we find that about 2.2 times more pairs
                        qualify for your expression than for Paul's. This is starting to
                        better approach the prime count difference for yours and Paul's
                        expressions.


                        Mark
                      • Nathan Russell
                        Sorry for replying to a very old thread, but I wondered if anyone has explored further with X^Y-Y^X (or for that matter X^Y+Y^X) in recent years. Some of the
                        Message 11 of 29 , Aug 13, 2009
                          Sorry for replying to a very old thread, but I wondered if anyone has
                          explored further with X^Y-Y^X (or for that matter X^Y+Y^X) in recent
                          years. Some of the PRPs listed are within easy reach of Primo, and
                          I'd be interested in working on some of them, but just want to know
                          which are already done.

                          Nathan
                        • cino hilliard
                          An anecdotal remark. I don t see the no use in USA warning. Can this software be used in USA now? http://www.ellipsa.eu/public/misc/license.html Cino To:
                          Message 12 of 29 , Aug 13, 2009
                            An anecdotal remark.



                            I don't see the no use in USA warning. Can this software be used in USA now?



                            http://www.ellipsa.eu/public/misc/license.html


                            Cino





                            To: primenumbers@yahoogroups.com
                            From: windrunner@...
                            Date: Thu, 13 Aug 2009 11:46:00 -0400
                            Subject: Re: [PrimeNumbers] Re: x^y - y^x






                            Sorry for replying to a very old thread, but I wondered if anyone has
                            explored further with X^Y-Y^X (or for that matter X^Y+Y^X) in recent
                            years. Some of the PRPs listed are within easy reach of Primo, and
                            I'd be interested in working on some of them, but just want to know
                            which are already done.

                            Nathan









                            [Non-text portions of this message have been removed]
                          • Norman Luhn
                            Hello, I think you can use PRIMO. Look here : http://www.sourcecodeonline.com/details/primo.html best Norman [Non-text portions of this message have been
                            Message 13 of 29 , Aug 13, 2009
                              Hello, I think you can use PRIMO.

                              Look here :

                              http://www.sourcecodeonline.com/details/primo.html

















                              best

                              Norman






                              [Non-text portions of this message have been removed]
                            • Nathan Russell
                              It looks from the homepage like it can be used in the US, just you are at your own risk for patents (and I think the chance of someone going after me is
                              Message 14 of 29 , Aug 13, 2009
                                It looks from the homepage like it can be used in the US, just you are
                                at your own risk for patents (and I think the chance of someone going
                                after me is approximately zero). Marcel also told me years ago that
                                as a paid user of the early versions, I had his permission to do so
                                even before the license change.

                                Nathan

                                On Thu, Aug 13, 2009 at 12:31 PM, Norman Luhn<nluhn@...> wrote:
                                >
                                >
                                > Hello, I think you can use PRIMO.
                                >
                                > Look here :
                                >
                                > http://www.sourcecodeonline.com/details/primo.html
                                >
                                >
                                >
                                >
                                >
                                >
                                >
                                >
                                >
                                > best
                                >
                                > Norman
                                >
                                > [Non-text portions of this message have been removed]
                                >
                                >
                              • Nathan Russell
                                ... (snip) ... These four, at least, are prime. ... I am working on the above 3 now. Should probably just go ahead and do the last - it should take under a
                                Message 15 of 29 , Aug 13, 2009
                                  2005/4/24 Décio Luiz Gazzoni Filho <decio@...>:
                                  > Paul Leyland maintains a table of primes of the form x^y + y^x. Has anyone
                                  > looked into the form x^y - y^x, and if not, is there any reason or has it
                                  > just been overlooked?
                                  >
                                  > I gave a bit of thought to them, and other than the condition gcd(x,y) = 1,
                                  > which was already known for x^y + y^x, I only came up with the following:
                                  > x^y
                                  > - y^x = -(y^x - x^y), and in particular x^y - y^x is positive if x < y (the
                                  > sole exceptions are the trivial x = 1 ones and 2^3 - 3^2 = -1), so those two
                                  > conditions cut the search space by 6/pi^2 = 0.61 and 0.5 respectively.
                                  >
                                  > I have exhaustively searched the range 1 < x < y <= 500 using a GP script
                                  > with
                                  > ispseudoprime() calls, and later proving the primality of PRPs using PRIMO.
                                  > I'm currently using PFGW with the -f switch to search the range 1 < x,500 <
                                  > y
                                  > <= 2000. I can't search far beyond this without a proper sieve. Perhaps Mark
                                  > Rodenkirch could patch MultiSieve to sieve this form? Since it already
                                  > supports x^y + y^x I guess the changes involved would be trivial.
                                  >
                                  > Here are the proven primes:
                                  (snip)
                                  > 42^425 - 425^42
                                  > 24^427 - 427^24
                                  >
                                  > Here are the pseudoprimes:
                                  >
                                  > 174^439 - 439^174
                                  > 42^445 - 445^42
                                  > 2^455 - 455^2
                                  > 2^461 - 461^2

                                  These four, at least, are prime.
                                  > 101^462 - 462^101
                                  > 172^471 - 471^172
                                  > 24^481 - 481^24

                                  I am working on the above 3 now. Should probably just go ahead and do
                                  the last - it should take under a minute.

                                  Nathan

                                  > 83^486 - 486^83
                                  > 384^487 - 487^384
                                  > 121^488 - 488^121
                                  > 173^488 - 488^173
                                  > 114^491 - 491^114
                                  > 329^494 - 494^329
                                  > 240^497 - 497^240
                                  > 455^498 - 498^455
                                  > 50^501 - 501^50
                                  > 3^512 - 512^3
                                  > 294^515 - 515^294
                                  > 485^516 - 516^485
                                  > 299^518 - 518^299
                                  > 390^527 - 527^390
                                  > 363^538 - 538^363
                                  > 398^539 - 539^398
                                  > 182^547 - 547^182
                                  > 475^552 - 552^475
                                  > 152^559 - 559^152
                                  > 360^559 - 559^360
                                  > 398^563 - 563^398
                                  > 14^579 - 579^14
                                  > 244^579 - 579^244
                                  > 323^584 - 584^323
                                  > 272^587 - 587^272
                                  > 332^589 - 589^332
                                  > 232^597 - 597^232
                                  > 323^606 - 606^323
                                  > 392^607 - 607^392
                                  > 215^608 - 608^215
                                  > 497^608 - 608^497
                                  > 2^623 - 623^2
                                  > 30^631 - 631^30
                                  > 532^633 - 633^532
                                  > 535^636 - 636^535
                                  > 302^641 - 641^302
                                  > 362^653 - 653^362
                                  > 14^655 - 655^14
                                  > 48^661 - 661^48
                                  > 119^668 - 668^119
                                  > 305^668 - 668^305
                                  > 629^668 - 668^629
                                  > 248^669 - 669^248
                                  > 116^675 - 675^116
                                  > 133^680 - 680^133
                                  > 301^692 - 692^301
                                  > 201^694 - 694^201
                                  > 171^704 - 704^171
                                  > 62^707 - 707^62
                                  > 446^715 - 715^446
                                  > 584^715 - 715^584
                                  > 49^720 - 720^49
                                  > 38^721 - 721^38
                                  > 240^721 - 721^240
                                  > 389^722 - 722^389
                                  > 81^724 - 724^81
                                  > 181^732 - 732^181
                                  > 38^735 - 735^38
                                  > 233^740 - 740^233
                                  > 398^741 - 741^398
                                  > 722^745 - 745^722
                                  > 649^774 - 774^649
                                  > 685^774 - 774^685
                                  > 691^774 - 774^691
                                  > 482^775 - 775^482
                                  > 113^782 - 782^113
                                  > 37^786 - 786^37
                                  > 334^789 - 789^334
                                  > 159^794 - 794^159
                                  > 325^798 - 798^325
                                  > 428^809 - 809^428
                                  > 21^814 - 814^21
                                  > 18^821 - 821^18
                                  > 282^821 - 821^282
                                  > 139^824 - 824^139
                                  > 12^833 - 833^12
                                  > 747^842 - 842^747
                                  > 338^859 - 859^338
                                  > 509^864 - 864^509
                                  > 181^878 - 878^181
                                  > 594^881 - 881^594
                                  > 883^884 - 884^883
                                  > 165^892 - 892^165
                                  > 472^897 - 897^472
                                  > 40^903 - 903^40
                                  > 272^903 - 903^272
                                  > 49^908 - 908^49
                                  > 398^913 - 913^398
                                  > 272^917 - 917^272
                                  > 866^925 - 925^866
                                  > 765^926 - 926^765
                                  > 847^930 - 930^847
                                  > 91^932 - 932^91
                                  > 740^933 - 933^740
                                  > 686^937 - 937^686
                                  > 702^941 - 941^702
                                  > 192^943 - 943^192
                                  > 770^943 - 943^770
                                  > 349^944 - 944^349
                                  > 373^944 - 944^373
                                  > 451^948 - 948^451
                                  > 841^950 - 950^841
                                  > 908^961 - 961^908
                                  > 291^962 - 962^291
                                  > 688^975 - 975^688
                                  > 8^977 - 977^8
                                  > 637^984 - 984^637
                                  > 38^985 - 985^38
                                  > 42^1003 - 1003^42
                                  > 849^1018 - 1018^849
                                  > 789^1030 - 1030^789
                                  > 408^1031 - 1031^408
                                  >
                                  > As anyone can check against Paul Leyland's tables at
                                  > http://www.leyland.vispa.com/numth/primes/xyyx.htm, the form x^y - y^x is
                                  > much more productive than x^y + y^x. For x,y < 1000, there are 254 x^y - y^x
                                  > primes and only 87 x^y + y^x primes. Frankly, this a mistery to a me. I see
                                  > no reason for them to be any different, even taking into account the fact
                                  > that x^y - y^x < x^y + y^x: the ratio x^y/y^x is always huge, even in the
                                  > most favorable case, x + 1 = y (here the ratio seems to be well approximated
                                  > by (x+y)/2e ~ 0.184(x+y)), so that the probability that each form is prime,
                                  > log(x^y + y^x) and log(x^y - y^x) are essentially the same.
                                  >
                                  > Interestingly, 2^9 + 9^2 and 2^9 - 9^2 is a `twin' pair, and the only one
                                  > I've
                                  > found yet. I'm not hoping to find a larger pair, but it would be a very
                                  > welcome surprise.
                                  >
                                  > Décio
                                  >
                                  >
                                  > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
                                  > The Prime Pages : http://www.primepages.org/
                                  >
                                  >
                                  >
                                  >
                                  > ________________________________
                                  > Yahoo! Groups Links
                                  >
                                  > To visit your group on the web, go to:
                                  > http://groups.yahoo.com/group/primenumbers/
                                  >
                                  > To unsubscribe from this group, send an email to:
                                  > primenumbers-unsubscribe@yahoogroups.com
                                  >
                                  > Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service.
                                • cino hilliard
                                  Thanks Norman and Nathan to all for verifying it. I downloaded it and will use it. I have to confess, I used it before too but the results did not go anywhere.
                                  Message 16 of 29 , Aug 13, 2009
                                    Thanks Norman and Nathan to all for verifying it.



                                    I downloaded it and will use it. I have to confess, I used it before too but

                                    the results did not go anywhere.



                                    Cino



                                    To: nluhn@...
                                    CC: primenumbers@yahoogroups.com
                                    From: windrunner@...
                                    Date: Thu, 13 Aug 2009 12:43:13 -0400
                                    Subject: Re: [PrimeNumbers] Re: x^y - y^x





                                    It looks from the homepage like it can be used in the US, just you are
                                    at your own risk for patents (and I think the chance of someone going
                                    after me is approximately zero). Marcel also told me years ago that
                                    as a paid user of the early versions, I had his permission to do so
                                    even before the license change.

                                    Nathan

                                    On Thu, Aug 13, 2009 at 12:31 PM, Norman Luhn<nluhn@...> wrote:
                                    >
                                    >
                                    > Hello, I think you can use PRIMO.
                                    >
                                    > Look here :
                                    >
                                    > http://www.sourcecodeonline.com/details/primo.html
                                    >
                                    >
                                    >
                                    >
                                    >
                                    >
                                    >
                                    >
                                    >
                                    > best
                                    >
                                    > Norman
                                    >
                                    > [Non-text portions of this message have been removed]
                                    >
                                    >









                                    [Non-text portions of this message have been removed]
                                  • Nathan Russell
                                    Just in case you re unaware, Primo, like any other non-special-form tool, is going to be much slower than those aimed at special forms this . For example,
                                    Message 17 of 29 , Aug 13, 2009
                                      Just in case you're unaware, Primo, like any other "non-special-form"
                                      tool, is going to be much slower than those aimed at special forms
                                      this . For example, the numbers I'm testing now (around 3 kbit) are
                                      taking about half an hour on my laptop. On the other hand, Mersenne
                                      3,217 takes under a second, and Mersenne 44,497, about the size of any
                                      number that might at all reasonably be ECPP tested in the next decade
                                      or so, takes under a minute.

                                      Sorry if you know this, or if that's not what you mean by "results not
                                      going anywhere".

                                      Nathan

                                      On Thu, Aug 13, 2009 at 12:53 PM, cino hilliard<hillcino368@...> wrote:
                                      >
                                      >
                                      > Thanks Norman and Nathan to all for verifying it.
                                      >
                                      > I downloaded it and will use it. I have to confess, I used it before too but
                                      >
                                      > the results did not go anywhere.
                                      >
                                      > Cino
                                      >
                                      >
                                      > To: nluhn@...
                                      > CC: primenumbers@yahoogroups.com
                                      > From: windrunner@...
                                      > Date: Thu, 13 Aug 2009 12:43:13 -0400
                                      > Subject: Re: [PrimeNumbers] Re: x^y - y^x
                                      >
                                      > It looks from the homepage like it can be used in the US, just you are
                                      > at your own risk for patents (and I think the chance of someone going
                                      > after me is approximately zero). Marcel also told me years ago that
                                      > as a paid user of the early versions, I had his permission to do so
                                      > even before the license change.
                                      >
                                      > Nathan
                                      >
                                      > On Thu, Aug 13, 2009 at 12:31 PM, Norman Luhn<nluhn@...> wrote:
                                      >>
                                      >>
                                      >> Hello, I think you can use PRIMO.
                                      >>
                                      >> Look here :
                                      >>
                                      >> http://www.sourcecodeonline.com/details/primo.html
                                      >>
                                      >>
                                      >>
                                      >>
                                      >>
                                      >>
                                      >>
                                      >>
                                      >>
                                      >> best
                                      >>
                                      >> Norman
                                      >>
                                      >> [Non-text portions of this message have been removed]
                                      >>
                                      >>
                                      >
                                      > [Non-text portions of this message have been removed]
                                      >
                                      >
                                    • Nathan Russell
                                      As a trivial observation, where x is not a power of 10, x^y-y^x has, with high probability, floor(y log x) digits. This is useful as a rule of thumb for how
                                      Message 18 of 29 , Aug 13, 2009
                                        As a trivial observation, where x is not a power of 10, x^y-y^x has,
                                        with high probability, floor(y log x) digits. This is useful as a
                                        rule of thumb for how long proving the numbers will take.

                                        Nathan
                                      • Mark Underwood
                                        ... The 101^462 - 462^101 passes pari s isprime() test. I don t know if that is foolproof however. What is curious to me is this: if the above is prime, it is
                                        Message 19 of 29 , Aug 14, 2009
                                          --- In primenumbers@yahoogroups.com, Nathan Russell <windrunner@...> wrote:
                                          > 101^462 - 462^101
                                          > 172^471 - 471^172
                                          > 24^481 - 481^24
                                          >
                                          > I am working on the above 3 now. Should probably just go ahead and > do
                                          > the last - it should take under a minute.
                                          >

                                          The 101^462 - 462^101 passes pari's isprime() test. I don't know if that is foolproof however.

                                          What is curious to me is this: if the above is prime, it is the first occurrence when an even multiple of 11 (462) has been noted.

                                          To my knowledge the numbers 22, 44, 66, etc. have not yet appeared to generated primes of the form x^y - y^x.

                                          Maybe numbers with factors of 11 have an almost complete covering set or something. Haven't checked.

                                          But I know why even squares > 4 (16,36,64,100,...) can never generate primes. :)


                                          Mark
                                        • Nathan Russell
                                          On Fri, Aug 14, 2009 at 3:08 PM, Mark ... It indeed is prime, one of about a dozen of this form I ve tested in the past day (all from the email I quoted).
                                          Message 20 of 29 , Aug 14, 2009
                                            On Fri, Aug 14, 2009 at 3:08 PM, Mark
                                            Underwood<mark.underwood@...> wrote:
                                            >
                                            >
                                            > --- In primenumbers@yahoogroups.com, Nathan Russell <windrunner@...> wrote:
                                            >> 101^462 - 462^101
                                            >> 172^471 - 471^172
                                            >> 24^481 - 481^24
                                            >>
                                            >> I am working on the above 3 now. Should probably just go ahead and > do
                                            >> the last - it should take under a minute.
                                            >>
                                            >
                                            > The 101^462 - 462^101 passes pari's isprime() test. I don't know if that is
                                            > foolproof however.
                                            >
                                            > What is curious to me is this: if the above is prime, it is the first
                                            > occurrence when an even multiple of 11 (462) has been noted.

                                            It indeed is prime, one of about a dozen of this form I've tested in
                                            the past day (all from the email I quoted).

                                            Nathan
                                          • Mark Underwood
                                            ... As it turns out I was seaching the wrong (and very incomplete) list! Lots of other even numbers with factors of 11 were on it. 44,66,88,110... Sorry about
                                            Message 21 of 29 , Aug 14, 2009
                                              --- In primenumbers@yahoogroups.com, Nathan Russell <windrunner@...> wrote:
                                              >
                                              > On Fri, Aug 14, 2009 at 3:08 PM, Mark
                                              > Underwood<mark.underwood@...> wrote:
                                              > >
                                              > > The 101^462 - 462^101 passes pari's isprime() test. I don't know if that is
                                              > > foolproof however.
                                              > >
                                              > > What is curious to me is this: if the above is prime, it is the first
                                              > > occurrence when an even multiple of 11 (462) has been noted.
                                              >
                                              > It indeed is prime, one of about a dozen of this form I've tested in
                                              > the past day (all from the email I quoted).
                                              >

                                              As it turns out I was seaching the wrong (and very incomplete) list! Lots of other even numbers with factors of 11 were on it. 44,66,88,110...
                                              Sorry about that.
                                              Mark
                                            • Andrey Kulsha
                                              ... http://xyyxf.at.tut.by/primes.html ... Best regards, Andrey [Non-text portions of this message have been removed]
                                              Message 22 of 29 , Aug 15, 2009
                                                > (or for that matter X^Y+Y^X)

                                                http://xyyxf.at.tut.by/primes.html

                                                :-)

                                                Best regards,

                                                Andrey

                                                [Non-text portions of this message have been removed]
                                              • Nathan Russell
                                                ... Awesome, thank you - somehow my google search missed that site. I may be masochistic, but would it be possible to reserve 1588^721+721^1588? Assuming
                                                Message 23 of 29 , Aug 15, 2009
                                                  On Sun, Aug 16, 2009 at 12:50 AM, Andrey Kulsha<Andrey_601@...> wrote:
                                                  >
                                                  >
                                                  >> (or for that matter X^Y+Y^X)
                                                  >
                                                  > http://xyyxf.at.tut.by/primes.html
                                                  >
                                                  > :-)
                                                  >
                                                  > Best regards,
                                                  >
                                                  > Andrey

                                                  Awesome, thank you - somehow my google search missed that site.

                                                  I may be masochistic, but would it be possible to reserve
                                                  1588^721+721^1588? Assuming Primo is O(log^4.5 n), which is my old
                                                  rule of thumb, I should be able to do this in about a month on my
                                                  machine.

                                                  Thanks,
                                                  Nathan
                                                • Nathan Russell
                                                  ... Where n is magnitude - somehow it slipped my mind that s the standard when talking about factoring algorithms, but not so much primality testing. Nathan
                                                  Message 24 of 29 , Aug 15, 2009
                                                    On Sun, Aug 16, 2009 at 1:32 AM, Nathan Russell<windrunner@...> wrote:

                                                    > I may be masochistic, but would it be possible to reserve
                                                    > 1588^721+721^1588?  Assuming Primo is O(log^4.5 n), which is my old
                                                    > rule of thumb, I should be able to do this in about a month on my
                                                    > machine.

                                                    Where n is magnitude - somehow it slipped my mind that's the standard
                                                    when talking about factoring algorithms, but not so much primality
                                                    testing.

                                                    Nathan
                                                  • Andrey Kulsha
                                                    ... OK :-) I ll update the pages next week. Thanks a lot, Andrey [Non-text portions of this message have been removed]
                                                    Message 25 of 29 , Aug 15, 2009
                                                      > I may be masochistic, but would it be possible to reserve
                                                      > 1588^721+721^1588? Assuming Primo is O(log^4.5 n), which is my old
                                                      > rule of thumb, I should be able to do this in about a month on my
                                                      > machine.

                                                      OK :-)

                                                      I'll update the pages next week.

                                                      Thanks a lot,

                                                      Andrey

                                                      [Non-text portions of this message have been removed]
                                                    • Nathan Russell
                                                      ... And done.  Andrey or anyone else want a copy of the certificate? Nathan
                                                      Message 26 of 29 , Sep 29, 2009
                                                        On Sun, Aug 16, 2009 at 1:52 AM, Andrey Kulsha <Andrey_601@...> wrote:
                                                        >
                                                        >
                                                        > > I may be masochistic, but would it be possible to reserve
                                                        > > 1588^721+721^1588? Assuming Primo is O(log^4.5 n), which is my old
                                                        > > rule of thumb, I should be able to do this in about a month on my
                                                        > > machine.
                                                        >
                                                        > OK :-)
                                                        >
                                                        > I'll update the pages next week.
                                                        >
                                                        > Thanks a lot,
                                                        >
                                                        > Andrey
                                                        >

                                                        And done.  Andrey or anyone else want a copy of the certificate?

                                                        Nathan
                                                      • Robdine
                                                        Hi all I don t remember how to open Pari (.tar.-file) can somebody help? gr. Rob Binnekamp ... From: Nathan Russell To: Andrey Kulsha Cc:
                                                        Message 27 of 29 , Sep 30, 2009
                                                          Hi all

                                                          I don"t remember how to open Pari (.tar.-file)
                                                          can somebody help?
                                                          gr. Rob Binnekamp


                                                          ----- Original Message -----
                                                          From: Nathan Russell
                                                          To: Andrey Kulsha
                                                          Cc: PrimeNumbers@yahoogroups.com
                                                          Sent: Wednesday, September 30, 2009 7:15 AM
                                                          Subject: Re: [PrimeNumbers] Re: x^y - y^x


                                                          On Sun, Aug 16, 2009 at 1:52 AM, Andrey Kulsha <Andrey_601@...> wrote:
                                                          >
                                                          >
                                                          > > I may be masochistic, but would it be possible to reserve
                                                          > > 1588^721+721^1588? Assuming Primo is O(log^4.5 n), which is my old
                                                          > > rule of thumb, I should be able to do this in about a month on my
                                                          > > machine.
                                                          >
                                                          > OK :-)
                                                          >
                                                          > I'll update the pages next week.
                                                          >
                                                          > Thanks a lot,
                                                          >
                                                          > Andrey
                                                          >

                                                          And done. Andrey or anyone else want a copy of the certificate?

                                                          Nathan




                                                          [Non-text portions of this message have been removed]
                                                        • maximilian_hasler
                                                          ... on linux you decompress and extract using (replace PARI with the actual file name): tar xvfz PARI.tar.gz (or pari.tgz) (omit the v if you don t want to
                                                          Message 28 of 29 , Sep 30, 2009
                                                            --- In primenumbers@yahoogroups.com, "Robdine" <robdine@...> wrote:
                                                            >
                                                            > Hi all
                                                            >
                                                            > I don"t remember how to open Pari (.tar.-file)
                                                            > can somebody help?
                                                            > gr. Rob Binnekamp

                                                            on linux you decompress and extract using
                                                            (replace PARI with the actual file name):

                                                            tar xvfz PARI.tar.gz (or pari.tgz)

                                                            (omit the "v" if you don't want to see the list of files) or

                                                            tar xvfj PARI.tar.bz2

                                                            If it's already decompressed, use simply

                                                            tar xvf PARI.tar

                                                            (That said you should be able to click on the file if you see it in some file manager window.)
                                                            On Windows, WinRAR and WinZIP should be able to open these,
                                                            on recent versions of the Win "OS" it works "automagically" I think.

                                                            Maximilian
                                                          • djbroadhurst
                                                            ... The form x^y-y^x was recently completely factorized for 0
                                                            Message 29 of 29 , Mar 13, 2012
                                                              --- In primenumbers@yahoogroups.com,
                                                              Décio Luiz Gazzoni Filho <decio@...> asked:

                                                              > Has anyone looked into the form x^y - y^x, and if not,
                                                              > is there any reason or has it just been overlooked?

                                                              The form x^y-y^x was recently completely factorized for 0 < x < y < 101:
                                                              http://groups.yahoo.com/group/ggnfs/files/factortable_xy-yx_1_100.txt

                                                              David
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