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MATH 465作业代做、代写NUMBER THEORY作业、Java,Python,c/c++编程语言作业代写代做留学生Prolog|代写Python程序

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 + 7y2̸≡ 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.

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