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Re: [PrimeNumbers] Generalised factorial primes??

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  • Edwin Clark
    ... for a from 2 to 30 and n from 2 to 30 I get the following primes of the form (a!+n)/n. There is apparently no shortage. a,n,(a!+n)/n 2, 2, 2 3, 3, 3 3, 6,
    Message 1 of 3 , Dec 5, 2003
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      On Fri, 5 Dec 2003, julienbenney wrote:

      > Using the ordinary equipment on my old home computer, I found that (13!
      > + 2)/2 is prime.
      >
      > How many primes of the form (a! + n)/n [or (a! - n)/n] are actually
      > known? Do you think there might by many to be found??
      >

      for a from 2 to 30 and n from 2 to 30 I get the following primes of the
      form (a!+n)/n. There is apparently no shortage.

      a,n,(a!+n)/n

      2, 2, 2
      3, 3, 3
      3, 6, 2
      4, 2, 13
      4, 4, 7
      4, 6, 5
      4, 12, 3
      4, 24, 2
      5, 2, 61
      5, 3, 41
      5, 4, 31
      5, 10, 13
      5, 12, 11
      5, 20, 7
      5, 30, 5
      6, 3, 241
      6, 4, 181
      6, 10, 73
      6, 12, 61
      6, 18, 41
      6, 20, 37
      6, 24, 31
      7, 2, 2521
      7, 5, 1009
      7, 8, 631
      7, 12, 421
      7, 15, 337
      7, 18, 281
      7, 21, 241
      7, 24, 211
      7, 28, 181
      8, 2, 20161
      8, 3, 13441
      8, 9, 4481
      8, 12, 3361
      8, 15, 2689
      8, 16, 2521
      8, 20, 2017
      9, 5, 72577
      9, 8, 45361
      9, 12, 30241
      9, 18, 20161
      9, 24, 15121
      9, 27, 13441
      9, 30, 12097
      10, 6, 604801
      10, 8, 453601
      10, 14, 259201
      10, 15, 241921
      10, 21, 172801
      10, 24, 151201
      10, 27, 134401
      11, 3, 13305601
      11, 5, 7983361
      11, 6, 6652801
      11, 7, 5702401
      11, 10, 3991681
      11, 20, 1995841
      11, 28, 1425601
      12, 8, 59875201
      12, 10, 47900161
      12, 12, 39916801
      12, 16, 29937601
      12, 21, 22809601
      13, 2, 3113510401
      13, 4, 1556755201
      13, 6, 1037836801
      13, 11, 566092801
      13, 16, 389188801
      14, 5, 17435658241
      14, 6, 14529715201
      14, 8, 10897286401
      14, 16, 5448643201
      14, 25, 3487131649
      14, 28, 3113510401
      14, 30, 2905943041
      15, 7, 186810624001
      15, 10, 130767436801
      15, 24, 54486432001
      16, 2, 10461394944001
      16, 7, 2988969984001
      16, 21, 996323328001
      16, 22, 951035904001
      17, 3, 118562476032001
      17, 6, 59281238016001
      17, 14, 25406244864001
      17, 24, 14820309504001
      17, 26, 13680285696001
      17, 27, 13173608448001
      18, 26, 246245142528001
      19, 5, 24329020081766401
      19, 15, 8109673360588801
      19, 20, 6082255020441601
      20, 8, 304112751022080001
      20, 30, 81096733605888001
      21, 4, 12772735542927360001
      21, 6, 8515157028618240001
      21, 11, 4644631106519040001
      21, 15, 3406062811447296001
      21, 19, 2688996956405760001
      22, 16, 70250045486100480001
      22, 25, 44960029111104307201
      22, 26, 43230797222215680001
      23, 3, 8617338912961658880001
      23, 5, 5170403347776995328001
      23, 8, 3231502092360622080001
      23, 10, 2585201673888497664001
      23, 15, 1723467782592331776001
      23, 24, 1077167364120207360001
      23, 27, 957482101440184320001
      23, 30, 861733891296165888001
      24, 8, 77556050216654929920001
      24, 15, 41363226782215962624001
      24, 22, 28202200078783610880001
      24, 27, 22979570434564423680001
      24, 28, 22158871490472837120001
      24, 30, 20681613391107981312001
      25, 4, 3877802510832746496000001
      25, 7, 2215887149047283712000001
      25, 26, 596585001666576384000001
      26, 10, 40329146112660563558400001
      28, 13, 23452949585516450807808000001
      28, 28, 10888869450418352160768000001
      29, 8, 1105220249217462744317952000001
      29, 12, 736813499478308496211968000001
      29, 15, 589450799582646796969574400001
      30, 2, 132626429906095529318154240000001

      trying again we have primes of the form (a!-n)/n

      a, n, (a!-n)/n
      3, 2, 2
      4, 2, 11
      4, 3, 7
      4, 4, 5
      4, 6, 3
      4, 8, 2
      5, 2, 59
      5, 4, 29
      5, 5, 23
      5, 6, 19
      5, 10, 11
      5, 15, 7
      5, 20, 5
      5, 30, 3
      6, 2, 359
      6, 3, 239
      6, 4, 179
      6, 8, 89
      6, 9, 79
      6, 10, 71
      6, 12, 59
      6, 15, 47
      6, 24, 29
      6, 30, 23
      7, 4, 1259
      7, 6, 839
      7, 7, 719
      7, 10, 503
      7, 12, 419
      7, 14, 359
      7, 20, 251
      7, 21, 239
      7, 28, 179
      7, 30, 167
      8, 4, 10079
      8, 6, 6719
      8, 8, 5039
      8, 12, 3359
      8, 14, 2879
      8, 15, 2687
      8, 18, 2239
      8, 28, 1439
      9, 2, 181439
      9, 7, 51839
      9, 14, 25919
      9, 16, 22679
      9, 20, 18143
      9, 28, 12959
      10, 4, 907199
      10, 8, 453599
      10, 12, 302399
      10, 15, 241919
      10, 16, 226799
      10, 18, 201599
      10, 20, 181439
      10, 27, 134399
      11, 5, 7983359
      11, 6, 6652799
      11, 8, 4989599
      11, 10, 3991679
      11, 12, 3326399
      11, 15, 2661119
      11, 30, 1330559
      12, 3, 159667199
      12, 5, 95800319
      12, 14, 34214399
      12, 28, 17107199
      13, 10, 622702079
      13, 13, 479001599
      13, 15, 415134719
      13, 30, 207567359
      14, 6, 14529715199
      14, 18, 4843238399
      14, 26, 3353011199
      15, 4, 326918591999
      15, 9, 145297151999
      15, 11, 118879487999
      15, 12, 108972863999
      15, 14, 93405311999
      15, 15, 87178291199
      15, 24, 54486431999
      15, 25, 52306974719
      16, 3, 6974263295999
      16, 5, 4184557977599
      16, 6, 3487131647999
      16, 8, 2615348735999
      16, 12, 1743565823999
      16, 15, 1394852659199
      16, 26, 804722687999
      17, 6, 59281238015999
      17, 9, 39520825343999
      17, 10, 35568742809599
      17, 25, 14227497123839
      18, 4, 1600593426431999
      18, 6, 1067062284287999
      18, 9, 711374856191999
      18, 14, 457312407551999
      18, 15, 426824913715199
      18, 16, 400148356607999
      18, 24, 266765571071999
      18, 30, 213412456857599
      19, 8, 15205637551103999
      19, 11, 11058645491711999
      19, 18, 6758061133823999
      19, 20, 6082255020441599
      19, 21, 5792623828991999
      19, 22, 5529322745855999
      19, 28, 4344467871743999
      20, 6, 405483668029439999
      20, 7, 347557429739519999
      20, 12, 202741834014719999
      20, 13, 187146308321279999
      20, 30, 81096733605887999
      21, 9, 5676771352412159999
      21, 14, 3649353012264959999
      21, 19, 2688996956405759999
      22, 25, 44960029111104307199
      23, 4, 6463004184721244159999
      23, 7, 3693145248412139519999
      23, 18, 1436223152160276479999
      23, 20, 1292600836944248831999
      23, 21, 1231048416137379839999
      23, 25, 1034080669555399065599
      23, 28, 923286312103034879999
      24, 28, 22158871490472837119999
      25, 12, 1292600836944248831999999
      25, 28, 553971787261820927999999
      26, 11, 36662860102418694143999999
      26, 22, 18331430051209347071999999
      26, 24, 16803810880275234815999999
      27, 9, 1209874383379816906751999999
      27, 23, 473429106539928354815999999
      28, 10, 30488834461171386050150399999
      28, 18, 16938241367317436694527999999
      28, 25, 12195533784468554420060159999
      29, 3, 2947253997913233984847871999999
      29, 9, 982417999304411328282623999999
      29, 13, 680135537979977073426431999999
      29, 18, 491208999652205664141311999999
      29, 28, 315777214062132212662271999999
      30, 9, 29472539979132339848478719999999
      30, 27, 9824179993044113282826239999999
      30, 29, 9146650338351415815045119999999

      And for a = 100 and n = 2 to 100 Maple finds the following primes
      of the form (a!+n)/n:

      a, n, (a!+n)/n

      100, 21,
      4444105497330673937223773278869842880510284203065791498504426852153219047296662648044831617912215156488271329563201131345294533853183999999999999999999999999
      100, 95,
      982381215199412133491570935329123163060168086993490752300978567318079999928735953778331199749015981960565241271865513244749318009651199999999999999999999999
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