1. Proposition statement: specify what the proposition is to be proved.

2. Assumptions or preconditions: Some assumptions or preconditions need to be relied upon in the process of proof and need to be clearly listed.

3. Proof process: Through logical reasoning and deduction, the correctness of the proposition is gradually proved. The proof process should be clear, rigorous and comprehensive, and there should be no omissions or omissions.

4. Conclusion: Summarize the proof results and make clear whether the conclusion is consistent with the proposition of the proof.

Mathematical proof model

Proposition: given a positive integer n, if n= 1(mod4) is satisfied, then n can be expressed as the sum of squares of two integers.

It is proved that let n be a positive integer and satisfy n= 1(mod4). Then, there must be positive integers x and y, so that n = x 2+y 2.

Suppose n= 1, then x= 1, y=0, which obviously satisfies n = x 2+y 2.

Suppose n> 1, then according to Fermat's sum of squares theorem, n can be expressed as the sum of squares of two integers if and only if all prime factors of n are powers of 4k+3.

Even multiple. Because n≡ 1(mod4), n can only have a prime factor in the form of 4k+ 1, and the power is even. Therefore, n can be expressed as the sum of squares of two integers.

Specifically, according to Lagrange theorem, any positive integer can be expressed as the sum of squares of four integers, that is, n = a 2+b 2+c 2+d 2. Because n= 1(mod4), at least one of the four integer representations of n must be an integer in the form of 4k+ 1.

Suppose a, n = a 2+b 2+c 2+d 2 can be simplified to n = a 2 (b 2+c 2+d 2). And b 2+c 2+d 2 is a positive integer, so n can be expressed as the sum of squares of two integers, that is, n = x 2+y 2, where x=a, y = √ (b 2+c 2+d 2), and both x and y are positive integers.