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Find all primes $p$ such that $\dfrac{(2^{p-1}-1)}{p}$ is a perfect square. I tried brute-force method and tried to find some pattern. I got $p=3,7$ as solutions . Apart from these I have tried for many other primes but couldn't find any other such prime. Are these the only primes that satisfy the condition ? .

If yes , how to prove it theoretically and if not, how to find others?.

Thanks in advance!!

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1 Answer 1

up vote 11 down vote accepted

Hint: Let $p=2k+1$ where $k \in \mathbb{N},$ then $2^{2k}-1=(2^k-1)(2^k+1)=pm^2.$ We have that $\gcd(2^k-1,2^k+1)=1$ since they are consecutive odd numbers, so the equation breaks into $2^k-1=px^2, 2^k+1=y^2$ or $2^k-1=x^2, 2^k+1=py^2.$ Easy investigation shows that the only solutions are $p=3,7.$ I will leave it to you to fill the gaps.

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Very good explanation.. +1. –  Iyengar Jun 22 '12 at 16:35
    
@ehsanmo: thanks.your hint really helped. –  Aang Jun 22 '12 at 18:32
    
Glad to hear that :) –  Ehsan M. Kermani Jun 22 '12 at 18:50
    
how to solve this equations? –  agustin Mar 8 '13 at 7:22
    
@Avatar how to solve this equations? with answers$3,7$ –  agustin Mar 9 '13 at 20:58

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