Find all positive integers x, y such that 2x+5y+2 is a perfect square.
Problem
Source: Kosovo MO 2020 Grade 11, Problem 2
Tags: number theory, Olympiad, Kosovo, Perfect Squares
02.02.2020 17:37
Circumcircle wrote: Find all natural numbers x, y such that 2x+5y+2 is a perfect square. Assuming that 0 is a natural number, here is my solution:
Just noticed the edit. So the first case can be ignored.
09.02.2020 11:49
Let 2^x+5^y+2=a^2If x\geq 2, LHS=RHS=a^2\equiv 3\pmod 4, which is impossible. So, x=1 5^y=a^2-4=(a-2)(a+2) gcd(a-2,a+2)=4, so both cannot be divisible by 5. Hence, a-2=1\implies a=3\implies y=1 Hence, the only solution is x=y=1
16.11.2020 19:36
Circumcircle wrote: Find all positive integers x, y such that 2^x+5^y+2 is a perfect square. For all x\geq 2 2^x+5^y+2\equiv 3\mod 4 which can never be perfect square. Hence x=1 5^y+4=z^2\implies 5^y=(z-2)(z+2)\implies z\equiv 2\mod 5, z\equiv -2\mod 5 Hence z-2=1, z+2=z^2-4\implies z=3 is only solution.
16.11.2020 19:50
Mm very clever...how did you know to take it mod 4 though?
17.11.2020 06:26
samrocksnature wrote: Mm very clever...how did you know to take it mod 4 though? Because 5^n \equiv 1 \pmod 4 and 2^n \equiv 0 \pmod 4 for n \ge 2, the expression for n \ge 2 will be congruent to 3 \pmod 4, which can never be a perfect square. This puts a very nice bound for n, and it's easy to check the case of n = 1. I'm kind of just restating what people before me have said, to be honest.
17.11.2020 16:44
Circumcircle wrote: Find all positive integers x, y such that 2^x+5^y+2 is a perfect square. First use mod 4 then you will get x can't be greater than 1. Then find out (x,y) if x=1