# The Distribution of Quadratic Residues

Theory of Numbers

The Distribution of Quadratic Residues

Sums of Squares

Sums of Four Squares

1. Let N1(p) denote the number of pairs of integers in [1, p - 1] in which the first is a quadratic residue

and the second is a quadratic nonresidue modulo p. Prove that

N1(p) = (1/4) (p - ( - 1)^((p - 1)/2))

2. Let N2(p) denote the number of pairs of integers in [1, p - 1] in which the first is a quadratic nonresidue

and the second is a quadratic residue modulo p. Prove that

N2(p) = (1/4) (p - 2 + ( - 1)^((p - 1)/2))

3. Let N3(p) denote the number of pairs of integers in [1, p - 1] in which the first is a quadratic nonresidue

and the second is a quadratic nonresidue modulo p. Prove that

N3(p) = (1/4) (p - 2 + ( - 1)^((p - 1)/2))

4. Use the results of theorem 10-4 and corollary 10-1 to construct solutions of x^2 + y^2 =29.

5. Prove ( without assuming corollary 10-1) that, if p is a prime ≡ 1 (mod 4), then there exists positive

integers m, x, and y such that x^2 + y^2 =mp, with p ┼ x, p ┼ y, 0 < m < p [ Hint: use the proof of

Theorem 11-2].

Theorem 10-4: If p is an odd prime, then ν(p) = (1/8)p + Ep where | Ep| < (1/4)(p)^(1/2) +2.

Corollary 10-1: Every prime p ≡ 1(mod 4) is representable as a sum of two squares.

Theorem 11-2: For each prime p there exist integers A,B and C, not all zero, such that

A^2 + B^2 + C^2 ≡ 0 (mod p).

See the attached file.

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Theory of Numbers

The Distribution of Quadratic Residues

Sums of Squares

Sums of Four Squares

1. Let N1(p) denote the number of pairs of integers in [1, p - ...

#### Solution Summary

This solution is comprised of a detailed explanation of the Distribution of Quadratic Residues

and Sums of Four Squares.

It contains step-by-step explanation for the following problem:

1. Let N1(p) denote the number of pairs of integers in [1, p - 1] in which the first is a quadratic residue

and the second is a quadratic nonresidue modulo p. Prove that

N1(p) = (1/4) (p - ( - 1)^((p - 1)/2))

2. Let N2(p) denote the number of pairs of integers in [1, p - 1] in which the first is a quadratic nonresidue

and the second is a quadratic residue modulo p. Prove that

N2(p) = (1/4) (p - 2 + ( - 1)^((p - 1)/2))

3. Let N3(p) denote the number of pairs of integers in [1, p - 1] in which the first is a quadratic nonresidue

and the second is a quadratic nonresidue modulo p. Prove that

N3(p) = (1/4) (p - 2 + ( - 1)^((p - 1)/2))

4. Use the results of theorem 10-4 and corollary 10-1 to construct solutions of x^2 + y^2 =29.

5. Prove ( without assuming corollary 10-1) that, if p is a prime ≡ 1 (mod 4), then there exists positive

integers m, x, and y such that x^2 + y^2 =mp, with p ┼ x, p ┼ y, 0 < m < p [ Hint: use the proof of

Theorem 11-2].

Theorem 10-4: If p is an odd prime, then ν(p) = (1/8)p + Ep where | Ep| < (1/4)(p)^(1/2) +2.

Corollary 10-1: Every prime p ≡ 1(mod 4) is representable as a sum of two squares.

Theorem 11-2: For each prime p there exist integers A,B and C, not all zero, such that

A^2 + B^2 + C^2 ≡ 0 (mod p).

Solution contains detailed step-by-step explanation.