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a^2 + b^2 + c^2 = d^2 if ab is even or odd

Source: 1984 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p3

November 21, 2022

number theoryPerfect Square

Problem Statement

Let

a

and

b

be positive integers. Show that if

a \cdot b

is even, then there are positive integers

c

and

d

with

a^2 + b^2 + c^2 = d^2

; if, on the other hand,

a\cdot b

is odd, there are no such positive integers

c

and

d