- 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 - --- In primenumbers@yahoogroups.com,

Décio Luiz Gazzoni Filho <decio@...> asked:

> Has anyone looked into the form x^y - y^x, and if not,

The form x^y-y^x was recently completely factorized for 0 < x < y < 101:

> is there any reason or has it just been overlooked?

http://groups.yahoo.com/group/ggnfs/files/factortable_xy-yx_1_100.txt

David