Competition problem — Number theory

Library / /333 of 520

Number theory Difficulty 6.3 National olympiad Prove it

Theorem 3 Let $n=m^{2} n_{1}, \mu\left(n_{1}\right) \neq 0$ , then the positive integer $n$ can be expressed as the sum of two squares of integers if and only if: whenever a prime $p \mid n_{1}$ , it must be that $p \equiv 1(\bmod 4)$ .

Solution

None

Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.

Note: The provided instruction is a meta-instruction and not part of the text to be translated. Since the text to be translated is "None", the translation is also "None". Here is the formatted output as requested:

None

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.