Vipul: /* Facts */

2009-05-05T16:07:58Z

Facts

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	* [[Smallest quadratic nonresidue is less than squareroot plus one]]: This states that the smallest quadratic nonresidue modulo <math>p</math> is less than <math>\sqrt{p} + 1</math>. This has been proved.		* [[Smallest quadratic nonresidue is less than squareroot plus one]]: This states that the smallest quadratic nonresidue modulo <math>p</math> is less than <math>\sqrt{p} + 1</math>. This has been proved.
	* [[Every integer that is not a perfect square is a quadratic nonresidue for infinitely many primes]]~~: In particular, this implies that the smallest quadratic nonresidue takes small values for arbitrarily large primes~~.		* [[Every prime is the smallest quadratic nonresidue for infinitely many primes]]: This follows from the fact that [[every integer that is not a perfect square is a quadratic nonresidue for infinitely many primes]].

Vipul: Created page with '==Definition== Let $p$ be a prime number. The '''smallest quadratic nonresidue''' modulo $p$ is the smallest positive integer $q$ such that ...'

2009-05-05T16:01:05Z

Created page with '==Definition== Let <math>p</math> be a prime number. The '''smallest quadratic nonresidue''' modulo <math>p</math> is the smallest positive integer <math>q</math> such that ...'

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==Definition==

Let <math>p</math> be a [[prime number]]. The '''smallest quadratic nonresidue''' modulo <math>p</math> is the smallest positive integer <math>q</math> such that the congruence class of <math>q</math> modulo <math>p</math> is not a square; in other words, <math>q</math> is the smallest [[defining ingredient::quadratic nonresidue]] modulo <math>p</math>.

The smallest quadratic nonresidue modulo a prime is always a prime.

==Facts and conjectures==

* [[Extended Riemann hypothesis]]: This states that the smallest quadratic nonresidue modulo <math>p</math> is less than <math>3 (\log p)^2/2</math>. This is a conjecture, and has not been proved.

===Facts===

* [[Smallest quadratic nonresidue is less than squareroot plus one]]: This states that the smallest quadratic nonresidue modulo <math>p</math> is less than <math>\sqrt{p} + 1</math>. This has been proved.
* [[Every integer that is not a perfect square is a quadratic nonresidue for infinitely many primes]]: In particular, this implies that the smallest quadratic nonresidue takes small values for arbitrarily large primes.

Smallest quadratic nonresidue - Revision history

Vipul: /* Facts */

Vipul: Created page with '==Definition== Let p be a prime number. The '''smallest quadratic nonresidue''' modulo p is the smallest positive integer q such that ...'

Vipul: Created page with '==Definition== Let $p$ be a prime number. The '''smallest quadratic nonresidue''' modulo $p$ is the smallest positive integer $q$ such that ...'