The goal of this problem is to classify the prime divisors of integers of the form n^(2)+n-3.
(a) Let p be a prime. Prove that 13 is a square modulo p if and only if p=2,p=13, or p is congruent to
1,3,4,9,10, or 12 modulo 13 .
(b) Prove that a prime p divides an integer of the form q(n)=n^(2)+n-3 if and only if p=13 or p is
congruent to 1,3,4,9,10, or 12 modulo 13. [Hint: What do you have to take the square root of?]
Part II: Solve the following problems. Justify all answers with rigorous, clear explanations.
4. The goal of this problem is to classify the prime divisors of integers of the form n2 + n -- 3.
(a) Let p be a prime. Prove that 13 is a square modulo p if and only if p = 2, p = 13, or p is congruent to 1,3,4,9,10,or 12 modulo 13.
(b) Prove that a prime p divides an integer of the form q(n) = n2 + n - 3 if and only if p = 13 or p is congruent to 1, 3, 4, 9, 10, or 12 modulo 13.[Hint: What do you have to take the square root of?