Let $x,y$ be two positive integers.
Is it possible for the expression $n=4xy-(x+y)$ to cover all the positive integers $n$ verifying $n=1(\mod 4)$.
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2026-02-22 17:59:44.1771783184
Is it possible for the expression $n=4xy-(x+y)$ to cover all the positive intergers $n$ verifying $n=1(mod4)$
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Take $n=4k+1$. We must prove that the equation $4xy-x-y=4k+1$ has integer answers. $$16xy-4x-4y=16k+4\\16xy-4x-4y+1=16k+5\\(4x-1)(4y-1)=16k+5$$ if $16k+1$ has a prime factor of form $4q-1$ the equation has such answers otherwise not. For example take $k=2$. There are no integer $x,y$ such that $(4x-1)(4y-1)=37$ or equivalently:$$4xy-x-y=9$$