Re: Triples correspond to sequences

From: Asger Grunnet (asger_at_adslhome.dk)
Date: 10/09/04 

Date: Sat, 9 Oct 2004 21:19:44 +0200

" Doug Goncz " wrote:
> For every triple of positive integers (a,b,c) there is associated a sequence {
> (a^n + b^n) mod c }.
>
> Is this a one-to-one correspondence?
>
> With the condition a < b < c < (a+b), is this one-to-one?
>
> With the addtional condition gcd(a,b,c), how about now?

I wrote a Maple program to check for triples with the same
associated sequence (using both of the extra conditions).

The first example it found was (3, 11, 12) and (3, 23, 24),
both of which give the sequence (2, 10, 2, 10, 2, 10, ...).
Another interesting example is (4, 7, 9) and (5, 29, 32), which
gives the sequence (2, 2, 2, ...).

Here's the Maple program:

S := [];
for a from 2 to 1000 do
  for b from a+1 to a+100 do
    for c from b+1 to a+b-1 do
      L := [];
      if gcd(a, gcd(b, c)) = 1 then
        for n from 1 to 200 do
          L := [op(L), modp(a^n + b^n, c)];
        end do;
        if member(L, S, i) then
          print({a, b, c}, S[i - 1], L);
        else
          S := [op(S), {a,b,c}, L];
        end if;
      end if;
    end do;
  end do;
end do;

Asger.