Does x 2 ≡ 3 (mod q) (where q is an odd prime) have infinite solutions?

Assume that q ≡ 1. The existence of infinite primes of this form is granted by Dirichlet's theorem. − 1 is a quadratic residue (mod ( mod q )) and by Cauchy's theorem there is an element with order 3 in Z / ( q Z ) ∗ ∗, which we may denote as ω. Since ω 2 + ω + 1 ≡ 0 ( mod q )we have ( 2 ω + 1 ) 2 + 3 ≡ 0 ( mod q )hence − 3 is a a quadratic residue (mod ( mod q )) and 3 is a quadratic residue as well.

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High School
Geometry
High school geometry
Indirect Proof

Kameron Wang

Answered question

2022-11-04

Does

x^{2} \equiv 3

(mod

q

) (where

q

is an odd prime) have infinite solutions?

Answer & Explanation

Prezrenjes0n

Beginner2022-11-05Added 19 answers

Assume that

q \equiv 1

. The existence of infinite primes of this form is granted by Dirichlet's theorem.

- 1

is a quadratic residue (mod

(\mod q)

) and by Cauchy's theorem there is an element with order

3

Z / (q Z)^{*}

∗, which we may denote as

ω

. Since

ω^{2} + ω + 1 \equiv 0 (\mod q)

we have

(2 ω + 1)^{2} + 3 \equiv 0 (\mod q)

hence

- 3

is a a quadratic residue (mod

(\mod q)

) and

3

is a quadratic residue as well.

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