Answer to Question #116847 in Discrete Mathematics for Pappu Kumar Gupta

Question

Write down the converse of each of the following statements: (2)
i) If n =1 (mod 4) for a natural number n, then n = x^2+y^2 for two integers
x and y.

Answer 1

Converse of given statement is

i) If there exists two integers $x$ and $y$ such that $n=x^2+y^2$ for a natural number $n$ , then $n \equiv 1(mod \ 4)$.

Note: This statement is not true in general.

E.g.: Let $8 = 2^2+2^2$ but 8 $\not\equiv 1(mod\ 4)$.