Congruent Products Under XOR; Fibbinary Numbers

[Dean Hickerson]

Paul D. Hanna wrote:

> As a generalization, consider the sequences defined by:
> "Numbers n such that: p*n XOR q*n = r*n."  for positive integers p, q, r.
>
> Q: For what p, q, r, does the above definition generate sequences not {0}?
> A: Iff  r = p XOR q (conjecture).
>
> Can anyone prove this conjecture?

It's not true.  Let  p=2,  q=3,  and  r=5.  Then  n=3  is in the sequence,
since  2*3 XOR 3*3 = 6 XOR 9 = 15 = 5*3.  But  2 XOR 3 = 1,  not 5.

Dean Hickerson
dean at math.ucdavis.edu