a's & b's

Number Theory Level 4

Two integers $a$ and $b$ satisfy $a+b=\sqrt{2(a^{2}+b^{2})-1}$ . Which of the following must be true?

The product of

a

and

b

must result in a perfect square. The sum of

a

and

b

must result in a perfect square

a

and

b

must be consecutive.

a

and

b

may not be zero

1 solution

Jesse Nieminen
Jul 7, 2016

Since $a+b > 0$ , we can square both sides.

$a+b = \sqrt{2(a^2+b^2) -1} \implies a^2 + 2ab + b^2 = 2a^2 + 2b^2 - 1 \implies (a-b)^2 = 1$

This implies that $a$ and $b$ must be consecutive.

Counter example can be quite easily found for other claims.

Hence the answer is $\boxed{\text{a and b must be consecutive.}}$

Nice solution

Biswajit Barik - 4 years, 3 months ago