- --- In primenumbers@yahoogroups.com, "David Broadhurst" <d.broadhurst@...> wrote:
>

For a prime p to occur twice with a gap in between then p must

> --- In primenumbers@yahoogroups.com,

> "Mike Oakes" <mikeoakes2@> wrote:

>

> > Every p appears in the divisor list either zero or one

> > or infinitely many times, and in the latter case at places

> > that are an arithmetic progression of n values.

>

> Very nice!

>

> Can you give an example where the arithmetic

> progression is different from the so-far-observed

> case of "every powering after the first appearance"?

divide x^m^n+c and x+c but not x^n+c.

It must therefore also divide the difference x^m^n-x but not x^n-x.

This is fine, however when we consider the next one x^m^n^n+c which it also must not divide for it to alternate or to occur in an arithmetic progression, then we're asking for it not to divide the difference x^m^n^n-x which is impossible as this is always divisible by x^m^n-x which we have agreed is divisible by p.

Complex behaviour can be constructed such as with 77783

3+77756 = 11 * 7069

3^3+77756 = 77783

3^3^3+77756 = 31 * 245987018153

3^3^3^3+77756 has factors 31 and 194819

3^3^3^3^3+77756 has factors 31, 13697, 31013

3^3^3^3^3^3+77756 has factors 31, 13697, 31013, 77783

3^3^3^3^3^3^3+77756 has factors 31, 13697, 31013, 77783 and these continue.

So the rules seem to be that a prime can occur once in isolation but that if it occurs again, it's in every subsequent time.

Richard Heylen - 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

https://groups.yahoo.com/neo/groups/primenumbers/conversations/topics/20526

:

"

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" ?

Thanks in advance!

Maximilian

--- In primenumbers@yahoogroups.com, "David Broadhurst"

<d.broadhurst@...> wrote:

> fail

(...)

I consider that his code should not be trusted.

David