Fermat Quotient of 2 wrt p is Square iff p is 3 or 7

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Theorem

Let $p$ be a prime number.

The Fermat quotient of $2$ with respect to $p$:

$\map {q_p} 2 = \dfrac {2^{p - 1} - 1} p$

is a square if and only if $p = 3$ or $p = 7$.


Proof

When $p = 3$:

$\map {q_3} 2 = \dfrac {2^{3 - 1} - 1} 3 = 1$

which is square.

When $p = 7$:

$\map {q_7} 2 = \dfrac {2^{7 - 1} - 1} 7 = \dfrac {63} 7 = 9$

which is square.



Sources