Difference of Squares and the Existence of Integers
Are there non-negative integers , , that satisfy ?
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1 solution
Subtracting b 2 from both sides and factoring the difference of squares gives ( a + b ) ( a − b ) = 2 × 3 c . The right-hand side is even, but not divisible by 4, so one of the factors ( a + b ) or ( a − b ) needs to be even and the other odd (otherwise their product would be divisible by 4). The sum of an even and an odd number is odd, but the sum ( a + b ) + ( a − b ) = 2 a is even. Hence, by contradiction, there are no such integers.