# Relate Mersenne Primes To Sq Triangular Nos.

1. May 16, 2006

### ramsey2879

Conjecture, For p>2, the $$2^{(p-1)}$$ th square triangular number is divisible by $$M_{p}$$ if and only if $$M_{p}$$ is prime. I checked this for 2<p<27. For instance the first four square triangular numbers are 0,1,36 and 1225 and the fourth is divisible by .
PS In fact it appears that if is prime then for any starting integers $$S_{1}$$ and $$S_2$$ having the recursive relation ,
$$S_{n} = 6*S_{n-1} - S_{n-2}$$ the following congruence holds:
$$S_{2^{p-1}} = S_{1} \mod M_{p}$$. There is prize money lurking here for those who are interested.

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