Solutions for $2^n+1=p^q$
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I have a problem I can’t solve, please help! Find all positive integer triples $(n,p,q)$ satisfying $2^n+1=p^q$ , where $p,q>1$ . There is a similar problem I can solve: Prove that it is not possible that $2^n-1=p^q$ , if $p,q>1$ . My solution: We need to prove that $2^n=p^q+1$ is not possible. Note that $p$ is an odd number and if you check $mod 4$ then you find that $q$ is also an odd number. Then $2^n=p^q+1=p^q+1^q=(p+1)(p^{q-1}-p^{q-2}+dots -p+1)$ . Note that $2^n$ doesn’t have an odd divisor $>1$ , but since $(p^{q-1}-p^{q-2}+dots -p+1)$ is an odd number $>1$ , contradiction.
number-theory elementary-number-theory diophantine-equations
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