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I've put on the http://mersenneforum.org/showthread.php?t=10670" the description of 3 conjectures that are waiting for a proof. I've already done half the proof for one of them (the easy part...).

I've provided PARI/gp code that exercises the 3 conjectures.

I'll give 100Euro for the guy/lady who will provide one proof out of the three.

These conjectures deal with properties of a Digraph under x^2-2 modulo a Mersenne, a Wagstaff or a Fermat prime, and they use cycles of length q-1 (Mersenne and Wagstaff) or 2^n-1 (Fermat).

The final idea/dream is to use cycles of length (q-1)/n for proving that such numbers are NOT prime, in order to speed up the search and proof.

The conjecture about Wagstaff numbers is already implemented as a "Vrba-Reix PRP" test in the LLR tool by Jean Penné, based on Prime95 library, which is usually used for looking at primes of the form: k*2^n+/-1 .

Come and have a look, have fun, and compete !

(And come look how big are the two new Prime Mersenne numbers we found at http://www.mersenne.org/" [Broken], and that I've verified.)

Tony

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# Three conjectures looking for a proof ! 100Euro reward !

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