Finding primes for which a given number is a perfect square.

Question

Find all primes p such that $(2^{p-1}-1)/p$ is a perfect square.

I know we can factorise $(2^{p-1}-1)$ into two distinct factors which will be coprime and hence p can divide one of these factors only. How to move further?

Answer 1

As you noticed, this factors into

$$ \frac{2^\ell \pm 1}{p} (2^\ell \mp 1), \text{ where } \ell = \frac{p-1}{2}. $$

Notice that both factors are coprime. If $x=uv$ is a square, with $(u,v)=1$, then you can probably show that $u$ and $v$ must both be squares.

For which $\ell$ is $2^\ell\pm 1$ square? In other words, let's solve. $$ 2^\ell \pm 1 = u^2.$$ Unless $\ell=0$, the left hand side is odd, so $u=2u'+1$ must be odd. $$ 2^{\ell} \pm 1 = u^2 = (2u'+1)^2 = 4u'^2 + 4u' + 1,$$

1. Fist assume the sign on the left hand side is $-1$, then we would have that $2^{\ell} -2 = 4u'^2 + 4u'$. Considering this equation mod $4$ severely reduces possibilities: $2^\ell \equiv 2 \pmod 4$ implies $\ell=1$. and indeed $p=3$ is a solution.
2. If the sign is $+1$, on the other hand, this reduces to $$ 2^{\ell-2} = u' (u'+1),$$ so $u'$ and $u'+1$ are consecutive powers of two, again severely limiting the possibilities: we must have that $u'=1$, i.e $u=3$, i.e. $\ell=3$, i.e. $p=7$. And indeed, $p=7$ is a solution.

Answer 2

Note that :

$$\frac{(2^{p-1}-1)}{p}=\frac{(2^{\frac{p-1}{2}}-1)(2^{\frac{p-1}{2}}+1)}{p}$$

Let us define $q$ and $r$ as :

$q=2^{\frac{p-1}{2}}+1$ and $r=2^{\frac{p-1}{2}}-1$

Note that : $q=r+2$

Suppose that : $p>2$ and $p \mid q \Rightarrow q=k \cdot p$ for $k \geq 1$

Now let us consider the case when $k>1$ :

$$\frac{q \cdot r}{p}=k \cdot r \Rightarrow k \mid r \Rightarrow r=n\cdot k \Rightarrow k\cdot p=n \cdot k+2 \Rightarrow p=n+\frac{2}{k}$$

which is contradiction since $p$ is a prime number greater than $2$ .

This means that if $p \mid q$ then $p=q$ , similarly one can show that if $p \mid r$ then $p=r$ .

So , we have shown that :

$p=2^{\frac{p-1}{2}}+1$ or $p=2^{\frac{p-1}{2}}-1$

All possible solutions are primes $3,5,7$ since right hand sides of the equalities above are bigger than left hand sides for every prime greater than $7$ .

Simply check gives us solutions : $3$ and $7$ .