I want to check if $2021m-1$ is a perfect square.
Let's suppose that it is. Then there is an interger n such that $$n^2=2021m-1\Rightarrow n^2=43\cdot 47m-1$$ How can we continue?
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Reduce mod $,p=43,$ then apply the ubiquitous linked theorem in the dupes. – Bill Dubuque Jun 10 '21 at 06:37
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So we get $n^2=42\pmod 43$,right? @BillDubuque – Mary Star Jun 10 '21 at 06:47
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Yes, $,n^2\equiv -1\pmod{!43},$ so the linked theorem applies. – Bill Dubuque Jun 10 '21 at 08:05