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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 &#9532; x, p &#9532; y, 0 < m < p [ Hint: use the proof of
Theorem 11-2].

Theorem 10-4: If p is an odd prime, then &#957;(p) = (1/8)p + Ep where | Ep| < (1/4)(p)^(1/2) +2.

Corollary 10-1: Every prime p &#8801; 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 &#8801; 0 (mod p).

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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 &#8801; 1 (mod 4), then there exists positive
integers m, x, and y such that x^2 + y^2 =mp, with p &#9532; x, p &#9532; y, 0 < m < p [ Hint: use the proof of
Theorem 11-2].

Theorem 10-4: If p is an odd prime, then &#957;(p) = (1/8)p + Ep where | Ep| < (1/4)(p)^(1/2) +2.

Corollary 10-1: Every prime p &#8801; 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 &#8801; 0 (mod p).

Solution contains detailed step-by-step explanation.

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