Conjecture, For p>2, the [tex]2^(p-1)[/tex] th square triangular number is divisible by [tex]M_{p}[/tex] if and only if [tex]M_{p}[/tex] 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 .(adsbygoogle = window.adsbygoogle || []).push({});

PS In fact it appears that if is prime then for any starting integers [tex]S_{1}[/tex] and [tex]S_2[/tex] having the recursive relation ,

[tex]S_{n} = 6*S_{n-1} - S_{n-2}[/tex] the following congruence holds:

[tex]S_{2^{p-1}} = S_{1} \mod M_{p}. There is prize money lurking here for those who are interested.

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# Relate Mersenne Primes To Sq Triangular Nos.

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