Loading ...
Sorry, an error occurred while loading the content.

Generalised factorial primes??

Expand Messages
  • julienbenney
    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
    Message 1 of 3 , Dec 5, 2003
    • 0 Attachment
      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??
    • cino hilliard
      Hi all, I just joined the group. I found it searching for a solution to a prime problem I had and am trying to prove. This caught my interest so i though i
      Message 2 of 3 , Dec 5, 2003
      • 0 Attachment
        Hi all,
        I just joined the group. I found it searching for a solution to a prime
        problem I had and am
        trying to prove. This caught my interest so i though i would chime in.


        >From: "julienbenney" <jpbenney@...>
        >To: primenumbers@yahoogroups.com
        >Subject: [PrimeNumbers] Generalised factorial primes??
        >Date: Fri, 05 Dec 2003 11:32:54 -0000
        >
        >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??
        Yes Many possiblt infinite. But watch out

        From the Pari script below.
        Here is a list n =1,2..50 and n=1,2 for sums

        a n (a! + n)/n
        1,1,2,
        2,1,3,
        2,2,2,
        3,1,7,
        4,2,13,
        5,2,61,
        7,2,2521,
        8,2,20161,
        11,1,39916801,
        13,2,3113510401,
        16,2,10461394944001,
        27,1,10888869450418352160768000001,
        30,2,132626429906095529318154240000001,
        37,1,13763753091226345046315979581580902400000001,
        41,1,33452526613163807108170062053440751665152000000001,
        43,2,30207631531686917818677566034256998753632256000000001,
        49,2,304140932017133780436126081660647688443776415689605120000000001,

        Another list for n = 1,2..50 for diff
        a n (a! - n)/n
        3,1,5,
        3,2,2,
        4,1,23,
        4,2,11,
        5,2,59,
        6,1,719,
        6,2,359,
        7,1,5039,
        9,2,181439,
        12,1,479001599,
        14,1,87178291199,
        30,1,265252859812191058636308479999999,
        31,2,4111419327088961408862781439999999,
        32,1,263130836933693530167218012159999999,
        33,1,8683317618811886495518194401279999999,
        38,1,523022617466601111760007224100074291199999999,
        41,2,16726263306581903554085031026720375832575999999999,

        Pari Script.
        nfactp2d2(n,m) =
        {
        for(x=1,n,
        for(k=1,m,
        y=floor((x!+ k)/k);
        if(isprime(y),print(x","k","y","))
        )
        )
        }
        nfactm2d2(n,m) =
        {
        for(x=1,n,
        for(k=1,m,
        y=floor((x!- k)/k);
        if(isprime(y),print(x",",k","y","))
        )
        )
        }
        If you dont have Pari I recommend it over all the expensive programs
        Maple,Mathematica etc
        it is free and available at

        http://pari.math.u-bordeaux.fr/

        Also, it has prime proving capability isprime() in addition to the much
        faster ispseudoprime.
        The script language is c-like but much better than c in terms of use and of
        course number
        theory capability.

        I don't know what your system is but if it is windows, I recommend you
        download the binary
        executable Pari.exe that will build all the files and folders for you. You
        will need to modify the
        environment path to include c:\program files\pari; This will enable you to
        call gp.exe from other
        folders.

        I submitted some sequences to sloane's and referenced this email and your
        question.

        Have fun in Primelandia
        Cino

        Behind some primes are other primes
        with other primes behind um.
        And behind these primes
        are still more primes
        and so ad infinitum.

        _________________________________________________________________
        Our best dial-up offer is back. Get MSN Dial-up Internet Service for 6
        months @ $9.95/month now! http://join.msn.com/?page=dept/dialup
      • 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 3 of 3 , Dec 5, 2003
        • 0 Attachment
          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
        Your message has been successfully submitted and would be delivered to recipients shortly.