## Re: Small prime divisors of very large numbers

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• ... I can shed some light on this behaviour. For any prime q, define b(q,n) by the recurrence relation b(q,n+1)=q^b(q,n), with initial condition b(q,0)=1. [So
Message 1 of 70 , Jul 1, 2009
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>
>
> > some primes pop in for an iteration and go again
> > and some come in and stick around
>
> How to predict which sort of behaviour?
>
> For example the divisor
> 93408839 | 137^(137^(137^(137^(137^137)))) + 184
> pops in and then out:
>
> {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}
>
> for(k=5,7,print(pmod(vector(k,j,137),93408839)+184))
>
> Mod(40934825, 93408839)
> Mod(0, 93408839)
> Mod(10826080, 93408839)
>

I can shed some light on this behaviour.

For any prime q, define b(q,n) by the recurrence relation
b(q,n+1)=q^b(q,n), with initial condition b(q,0)=1.
[So b(q,1)=q, b(q,2)=q^q, and so on.]

Theorem: For k any integer, if a prime p divides (b(q,n)+k) and
(b(q(n+1)+k) then p divides (b(q,n+2)+k).
Proof: Assume the conditions of the theorem hold.
Then b(q,n+1) = b(q,n) mod p,
so q^b(q,n+1) = q^b(q,n) mod p,
i.e. b(q,n+2) = b(q,n+1) mod p,
so (b(q,n+2)+k) = (b(q,n+1)+k) mod p = 0.
Q.E.D.

In words: once any prime p has "popped in" to the divisor list for /two consecutive/ terms of the sequence, then it stays there for ever.

(So those 20 mins of computer time described in my post of yesterday were unnecessary, except as experimental confirmation.)

-Mike Oakes
• 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 5:29 PM
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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