isi-entrance 2021 Q22

isi-entrance · India · UGA Number Theory Quadratic Diophantine Equations and Perfect Squares

For a positive integer $n$, the equation $$x ^ { 2 } = n + y ^ { 2 } , \quad x , y \text { integers} ,$$ does not have a solution if and only if
(A) $n = 2$.
(B) $n$ is a prime number.
(C) $n$ is an odd number.
(D) $n$ is an even number not divisible by 4 .

For a positive integer $n$, the equation
$$x ^ { 2 } = n + y ^ { 2 } , \quad x , y \text { integers} ,$$
does not have a solution if and only if\\
(A) $n = 2$.\\
(B) $n$ is a prime number.\\
(C) $n$ is an odd number.\\
(D) $n$ is an even number not divisible by 4 .

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