Do it without a calculator. I double-dare you

Number Theory Level 5

If the least solution for the equation

$x^2 - 853777y^2 = 1$

can be represented as $(x, y)$ , where $x$ and $y$ are positive integers, find $x + y$ .

1 solution

Ronald Overwater
Sep 17, 2015

Rewriting the equation into: $(x-924y)(x+924y)=(1+y^2)$

Assuming $x= 924y +a$ , gives $a(a+1848y)= a^2 +1848 a y = 1 + y^2$

$a=1$ and $y=1848$ is a trivial solution.

$x= 924 \times 1848 +1 = 1707553$

$x+y = 1707553 + 1848 = \fbox{1709401}$