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Challenge 324: Reasoner's Return III

Dr Reasoner has another claim for you to investigate...

Dr Reasoner (see challenges 160, 173, 191, 206, amongst others) has been investigating some more claims about numbers.

From her investigations she is almost certain that there is only one pair of integers m and n for which the expression 2m + 3n is a perfect square.

But is this correct? Justify your answer clearly!