## Re: Small prime divisors of very large numbers

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• ... Indeed there is a nonsequitor, David :-( While I try and mend the proof (I still believe the Theorem to be true - HELP, anyone?), here are the results of a
Message 1 of 70 , Jul 2, 2009
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>
> "Mike Oakes" <mikeoakes2@> wrote:
>
> > Then b(q,n+1) = b(q,n) mod p,
> > so q^b(q,n+1) = q^b(q,n) mod p,
>
> Err, Mike, how does the second line follow from the first?
>
> If x = y mod (p-1)
> then q^x = q^y mod p,
> in my book.
>
> Has one of us made a big boob?

Indeed there is a nonsequitor, David :-(

While I try and mend the proof (I still believe the Theorem to be true - HELP, anyone?), here are the results of a nontrivial run that has just finished (17.2 hrs @3.79GHz).

I used Jens's pari script, with the prime limit upped from 10^8 to 4*10^9, which is just about as high as you can go with 32-bit gp.exe, on your original puzzle form of 137^^n+73.

Here are the sets of prime divisors up to that limit:-
n=1: 2, 3, 5, 7 [done manually, since 137+73=210]
n=2: 2, 3, 5, 821, 71757331, 152555243
n=3: 2, 3, 5, 29, 821
n=4: 2, 3, 5, 29, 821
n=5: 2, 3, 5, 29, 821, 23339, 67525153, 224354393
6<=n<=15: 2, 3, 5, 29, 821, 23339, 67525153

They still support my Theorem.
Can anyone find a counterexample to it, for any (q^^n+k sequence)?

Mike
• Dear all, I come back to this topic, looking for a working pow_mod . This : {mypmod(b,n,p)=local(m=[p],f=0); while(n=n-1,m=concat(eulerphi(m[1]),m));
Message 70 of 70 , Mar 23, 2014
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Dear all,
I come back to this topic, looking for a working "pow_mod".
This :
{mypmod(b,n,p)=local(m=[p],f=0);
while(n=n-1,m=concat(eulerphi(m[1]),m));
for(p=n=1,#m,n=lift(Mod(b,m[p])^n);
if(f=f+(n*log(b)>=log(m[p])),n=n+m[p]));n%m[#m];}

yields:
mypmod(2,3,5)
= 2
but 2^2^2 % 5 = 16 % 5 = 1.
(and idem for 2^2^2^2 - of course.).

David gave some other code, in
:
"
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.
(...)
{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}
"
Although this may work for prime m
(at least, pmod([2,2,2],5) = Mod(1,5) as should),
it gives:

pmod([2,2,2],4)
= Mod(1, 4)
which is most certainly wrong.

So, to put it short, has anyone a working pmod() in his "library" ?

Maximilian