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Challenge 324: Reasoner's Return III

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!

Submit your solution

Please do send in your solution to this problem to  weeklymaths@kcl.ac.uk  You can scan or photograph your written work, or type your solutions. If this is your first weekly maths challenge solution, please include your year group and the name of the school you attend. We'll be happy to provide feedback on your solution, assuming that you are in year 11 or below. If you are older than this, we hope you enjoy trying the problems and reviewing your solutions against those we publish on the website.