Two integers and satisfy . Which of the following must be true?
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Since a + b > 0 , we can square both sides.
a + b = 2 ( a 2 + b 2 ) − 1 ⟹ a 2 + 2 a b + b 2 = 2 a 2 + 2 b 2 − 1 ⟹ ( 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 a and b must be consecutive.