For each odd prime p, we consider the two numbersA = sum of all 1 ≤ a such that a is a qua...

For each odd prime p, we consider the two numbers

A = sum of all 1 ≤ a such that a is a quadratic residue modulo p,

B = sum of all 1 ≤ a such that a is a nonresidue modulo p.

For example, if p = 11, then the quadratic residues are

12 ≡ 1 (mod 11), 22 ≡ 4 (mod 11), 32 ≡ 9 (mod 11),

42 ≡ 5 (mod 11), 52 ≡ 3 (mod 11),

so

A = 1 + 4 + 9 + 5 + 3 = 22 and B = 2 + 6 + 7 + 8 + 10 = 33.

(a) Make a list of A and B for all odd primes p<20.

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(b) What is the value of A + B? Prove that your guess is correct.

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(c) Compute A mod p and B mod p. Find a pattern and prove that it is correct. [Hint. See Exercise 7.4 for a formula for 12 + 22 + • • • + n2 that might be useful.]

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(d) Compile some more data and give a criterion on p which ensures that A = B. After reading Chapter 21, you will be asked to prove your criterion.

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(e) Write a computer program to compute A and B, and use it to make a table for all odd p<100. If A ≠ B, which one tends to be larger, A or B? Try to prove that your guess is correct, but be forewarned that this is a very difficult problem.