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sqrtm-sqrtn =p implies m,n, are squares

Source: Romanian District Olympiad 2017, Grade VIII, Problem 1

October 10, 2018
number theoryeasy

Problem Statement

a) Let m,n,p∈Z≥0 m,n,p\in\mathbb{Z}_{\ge 0} such that m>n m>n and m−n=p. \sqrt{m} -\sqrt n=p. Prove that m m and n n are perfect squares.
b) Find the numbers of four digits abcd‾ \overline{abcd} that satisfy the equation: abcd‾−acd‾=bb‾. \sqrt {\overline{abcd} } -\sqrt{\overline{acd}} =\overline{bb} .
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