# Prove the sum of two even perfect squares is not a perfect square

by vinnie
Tags: perfect, prove, square, squares
 P: 23 1. The problem statement, all variables and given/known data For all natural numbers, a and b, if a and b are both even, then (a^2+b^2) is not a perfect square. (prove this) 2. Relevant equations 3. The attempt at a solution I tried proving by contradiction and got (2s)^2 +(2t)^2 =k^2. which translates to 4s^2 +4t^2=k^2. I don't know how to form the contradiction from here. Is it even possible?
 P: 371 $$(a+b)^2=a^2+2ab+b^2$$
Mentor
P: 15,166
 Quote by vinnie 1. The problem statement, all variables and given/known data For all natural numbers, a and b, if a and b are both even, then (a^2+b^2) is not a perfect square. (prove this)
Are you sure you have the problem correct as stated? As stated this is easily proven false by counterexample.

P: 23
Prove the sum of two even perfect squares is not a perfect square

 Quote by D H Are you sure you have the problem correct as stated? As stated this is easily proven false by counterexample.
so I set up the negation, then we assume a and b are even and that a^2 +b^2 is a perfect square. Then subbing 4 for a and 6 for b, we get a contradiction?
 Mentor P: 15,166 Obviously 52 is not a perfect square. The conjecture does not say that the sum of squares of some specific pair of even numbers is not a square number. The conjecture says that the sum of squares of every pair of even numbers is not a square number.
P: 23
 Quote by D H Obviously 52 is not a perfect square. The conjecture does not say that the sum of squares of some specific pair of even numbers is not a square number. The conjecture says that the sum of squares of every pair of even numbers is not a square number.
true, but using four and six as counterexamples.....
 Mentor P: 15,166 42+62=16+36=52. and 52 is not a perfect square. 4 and 6 do not form a counterexample.
P: 23
 Quote by D H 42+62=16+36=52. and 52 is not a perfect square. 4 and 6 do not form a counterexample.
a counterexample in the negation of the conjecture.
 Mentor P: 15,166 Correct, and 52 is not a perfect square. 4 and 6 are consistent with the conjecture.
 P: 23 we assume the negation of the conjecture. which is for all natural numbers, a and b, if a and b are both even, then (a^2 +b^2) IS a perfect square. if we use 4 and 6 as counterexamples we do not get a perfect square, so we have a contradiction.....
 P: 23 or are you saying we don't need the negation, just provide 4 and 6 as counterexamples and be finished....?
 P: 23 There is an error.
 P: 23 or maybe we're supposed to prove the conjecture false by counterexample... using 6 and 8 perhaps.