Need help to understand this solution of $2^mp^2 +1=q^5$

Question

The Question-

Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , > q $ are primes.

****$2^mp^2 +1=q^5$****

After trying my best to solve this problem, and progressing a little bit, I decided to look up the solution and I am unable to quite understand it.

The solution-

clearly $q,p$ are odd

Let:$A=\sum _{i=0}^4q^i$ ,

$\large 2^mp^2=q^5-1=(q-1)\left ( A \right )$ Let:$d=\gcd(q-1,A)$$\large \Rightarrow d|A=\left (\sum _{i=0}^4(q^i-1) > \right )+5\Leftrightarrow d|5$

If:$d=5\large \Rightarrow p=5\Rightarrow A=5$ since$ A$ is odd , But $A>1+1+1+1+1=5$

then $d = 1$ $\large \Rightarrow q-1=2^m,A=p^2$ , Assume that $ > m\geq3$ $\large \Rightarrow q\equiv1[8]\Rightarrow p^2=A\equiv5[8]$

Which is impossible , so $m<3$

If:$m=1 \Rightarrow q=3,p=11$ $m=2$ $\large \Rightarrow q=5\Rightarrow > p^2=781$

Therefore the unique solution is $(1,11,3)$

I have understood upto the part where $d$ divides 5, hence $d=1$ or $5$ but I don't understand how the solution progresses after that.

Could someone give me an explanation behind this proof, and also the thought process of the solution so that I can solve questions like this on my own

Answer 1

I'm going to add some words to the solution.

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$$2^mp^2=(q-1)A\tag1$$ where $A=q^4+q^3+q^2+q+1$.

If:$d=5\large \Rightarrow p=5\Rightarrow A=5$ since$ A$ is odd , But $A>1+1+1+1+1=5$

If $d=5$, then the right hand of $(1)$ is divisible by $5$, so it follows that $p=5$. Since $A$ is odd, we see that $2^m$ divides $q-1$. Noting that $5$ divides both $q-1$ and $A$, we see that $A=5$. However, this is impossible since $$A=q^4+q^3+q^2+q+1\gt 1+1+1+1+1=5$$

then $d = 1$ $\large \Rightarrow q-1=2^m,A=p^2$ , Assume that $m\geq3$ $\large \Rightarrow q\equiv1[8]\Rightarrow p^2=A\equiv5[8]$ which is impossible , so $m<3$

If:$m=1 \Rightarrow q=3,p=11$ $m=2$ $\large \Rightarrow q=5\Rightarrow p^2=781$

When $d=1$, since $A$ is odd, we see that $q-1=2^m$ and $A=p^2$.

Assume that $m\ge 3$. Then, $q=2^m+1\equiv 0+1\equiv 1\pmod 8$, so $$p^2=A=q^4+q^3+q^2+q+1\equiv 1^4+1^3+1^2+1+1\equiv 5\pmod 8$$ However, there is no integer $p$ satisfying $p^2\equiv 5\pmod 8$.

So, we get $m\lt 3$.

If $m=1$, then $q=2^1+1=3$ and $2p^2=3^5-1$ implies $p=11$.

If $m=2$, them $q=2^2+1=5$ and $4p^2=5^5-1$ implies $p=\sqrt {781}$ which is not an integer.

Answer 2

So the game is you want to look at divisibility and prime factorizations of both sides of $2^mp^2 = q^5 - 1$ using each of the assumptions.

Starting with where you say you're stuck:

First, if $d=5$, then the right hand side is divisible by 5, so the left hand side is as well. Since $p$ is an odd prime, this implies that $p=5$ (as otherwise 5 can't divide the left hand side). Now we known that $A$ is a divisor of the left hand side, $5|A$, and $A$ is odd. The only odd divisors of the left hand side are 5 and 25. But 5 only divides the left hand side twice, and by assumption divides both $q-1$ and $A$, so we must have that $A$ is 5. But $A$ is also the sum of 5 terms each $\geq 1$, and not all are equal to $1$, so on the other hand we have that $A>5$. This is a contradiction, so there are no cases where $d=5$.

The second part is more analysis of this type. Hopefully this helps!