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Algebra Level 3

Find all triples of positive integers $(x, y, z)$ satisfying $1+2^{x}3^{y}=z^{2}$ .

Insert your answer as the sum of all possible values of $z$ .

1 solution

X X
Aug 9, 2018

$2^x3^y=(z+1)(z-1)$ , $z+1$ and $z-1$ have to be even numbers,so let $z-1=2a,z+1=2b,x-2=x'$ .

$2^{x'}3^y=ab$ ,so one of $a$ or $b$ has to be a power of $2$ ,and the other has to be a power of $3$ .

Because of Mihailescu's theorem , $(a,b)=(2,3),(3,4),(8,9)$

$z=5,7,17$

You could quote Mihailescu's theorem to show that these are the only possible values for $a$ and $b$

Freddie Hand - 2 years, 10 months ago

Thanks.I've editted it.

X X - 2 years, 10 months ago