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AW: Fwd: [PrimeNumbers] Re: Small prime divisors of very large numbers

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  • Norman Luhn
    I m not sure I get only 5 factors. 2,3,5,11,821 . Note , I have also 821 ! No other found. Here my UBASIC program    10   P=1    20   P=nxtprm(P)
    Message 1 of 2 , Jul 1 6:27 AM
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      I'm not sure I get only 5 factors.

      2,3,5,11,821 . Note , I have also 821 ! No other found.

      Here my UBASIC program

         10   P=1
         20   P=nxtprm(P)
         30   S1=modpow(137,137,P-1)
         40   S2=modpow(137,S1,P-1)
         50   S3=modpow(137,S2,P-1)
         60   S4=modpow(137,S3,P-1)
         70   S5=modpow(137,S4,P)+73
         80   if S5@P=0 then print P;
         90   goto 20
      OK

      Hm ?

      > > The first 7 primes that divide

      > > 137^(137^(137^ (137^137) )) + 73

      > > are 2, 3, 5, 29, 821, 23339, 67525153.
    • David Broadhurst
      ... In line 50, use the modulus m3 = eulerphi(P-1). In line 40, use the modulus m4 = eulerphi(m3). In line 30, use the modulus m5 = eulerphi(m4). Below is a
      Message 2 of 2 , Jul 1 9:22 AM
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        --- In primenumbers@yahoogroups.com,
        Norman Luhn <nluhn@...> wrote:

        > I get only 5 factors.

        These lines were wrong:

        >    30   S1=modpow(137,137,P-1)
        >    40   S2=modpow(137,S1,P-1)
        >    50   S3=modpow(137,S2,P-1)

        In line 50, use the modulus m3 = eulerphi(P-1).
        In line 40, use the modulus m4 = eulerphi(m3).
        In line 30, use the modulus m5 = eulerphi(m4).

        Below is a Pari-GP procedure "pmod(a,m)" to compute
        a[1]^(a[2]^(a[3]^ ... ^(a[k-1]^a[k]) ... ) modulo m
        where the modulus "m" need not be prime.
        I shall use it to show that
        3^6 * 11 * 13^2 * 277 * 1063 * 7459 * 93408839
        divides
        137^(137^(137^(137^(137^137)))) + 184

        {pmod(a,m)=local(k,q,t);k=#a;q=[m];t=a[k];
        for(j=2,k-1,q=concat(eulerphi(q[1]),q));
        for(j=1,k-1,t=Mod(a[k-j],q[j])^lift(t));t}

        m = 3^6 * 11 * 13^2 * 277 * 1063 * 7459 * 93408839;
        print(pmod([137,137,137,137,137,137],m)+184)

        Mod(0, 278027998329615967980261)

        David
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