MATH 465代做、代写NUMBER THEORY、Java，Python，c/c++编程语言代写-代写Python编程

MATH 465 NUMBER THEORY, SPRING 2020, PROBLEMS 11

Return by Monday 6th April

1. (i) Let P denote the set of odd primes p for which ?7 is a quadratic residue

modulo p. Prove that p ∈ P if and only if p is odd and p ≡ 1, 2 or 4 (mod 7). Note

that if p ∈ P, then p > 7.

(ii) Prove that if p ∈ P, then there are x, y, m such that

x2 + 7y2 = mp

and 1 ≤ m ≤ 7.

(iii) Prove that m ?= 7.

(iv) Prove that if x and y are both odd, then x2 + 7y2 ≡ 0 (mod 8). Deduce

that in (ii) x and y cannot both be odd.

(v) Prove that if x and y are both even, then x2 + 7y2 ≡ 0 (mod 4) and so in

(ii) m would be 4. Deduce that (x/2)2 + 7(y/2)2 = p.

(vi) That leaves m = 1, 3 or 5 in (ii). Prove that if m = 3 or 5 in (ii), then

(m, xy) = 1.

(vii) Prove that if (3, xy) = 1, then x2 + 7y2≡ 0 (mod 3) and hence that in (ii)m ?= 3.

(vii) Prove that if (5, xy) = 1, then x2 + 7y2?≡ 0 (mod 5) and hence that in (ii)

m ?= 5.

(viii) Prove that p ∈ P if and only if p ?= 7 and there are x and y such that

x2 + 7y2 = p.

(ix) Prove that there are infinitely many primes in P.

This is a curious example. The discriminant of the form x2+7y2

is ?4.1.7 = ?28

and there is another “reduced” form 2x

2 + 2xy + 4y

2 with the same discriminant.

However this form represents only even numbers, so the only prime it can represent

is 2, which in some sense is one of the “missing” primes from P. It also “imprimitive”

in the sense that the coefficients have a common factor greater than 1. The

other missing prime is 7, which is represented by x

2 + 7y2.