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I'm looking to solutions of: [tex]2^n+Q=m^2[/tex] , where [tex]Q=1[/tex] .

Obviously, n must be odd.

I already know the trivial solution: [tex]2^3+1=3^2[/tex] and I've started using a naive PARI/gp program for finding (n,m) up to n=59 . No success yet.

Do you know about other solutions or about some theory ?

This is related to Pell numbers [tex](P,Q)=(2,-1)[/tex] and to a series of Prime numbers studied by Newman, Shanks and Williams, called NSW numbers, and generated by: [tex](P,Q)=(6,1)[/tex] .

The idea is to have [tex]D=P^2-4Q=2^n[/tex] and [tex]Q=\pm 1[/tex] .

Since Mersenne numbers are square-free, (2,-1) is the unic solution for Q=-1.

About Q=1, I don't know ...

Tony

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# 2^n+1 = m^2

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