Difference between revisions of "Smallest quadratic nonresidue is less than square root plus one"

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(Created page with '==Statement== Suppose <math>p</math> is an odd prime. Suppose <math>a</math> is the smallest positive integer less than <math>p</math> that is a quadratic nonresidue modulo ...')
 
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==Related facts==
 
==Related facts==
  
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===Stronger facts===
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* [[Subgroup of multiplicative group of prime field is generated by elements less than squareroot plus one]]
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===Other related facts and conjectures===
 
* [[Extended Riemann hypothesis]]: A stronger form of the Riemann hypothesis that is equivalent to the statement that the smallest quadratic nonresidue modulo <math>p</math>, for any odd prime <math>p</math>, is less than <math>3(\log p)^2/2</math>, where <math>\log</math> here is the natural logarithm.
 
* [[Extended Riemann hypothesis]]: A stronger form of the Riemann hypothesis that is equivalent to the statement that the smallest quadratic nonresidue modulo <math>p</math>, for any odd prime <math>p</math>, is less than <math>3(\log p)^2/2</math>, where <math>\log</math> here is the natural logarithm.
 
* [[Artin's conjecture on primitive roots]]: This is a closely related conjecture that states that every integer that is not <math>-1</math> or a perfect square occurs as a primitive root for infinitely many primes.
 
* [[Artin's conjecture on primitive roots]]: This is a closely related conjecture that states that every integer that is not <math>-1</math> or a perfect square occurs as a primitive root for infinitely many primes.

Revision as of 13:19, 21 April 2009

Statement

Suppose is an odd prime. Suppose is the smallest positive integer less than that is a quadratic nonresidue modulo . Then:

.

Note that we need the additive correction term. For instance, in the cases and , the smallest quadratic nonresidues ( and respectively) are greater than the squareroot.

Related facts

Stronger facts

Other related facts and conjectures

  • Extended Riemann hypothesis: A stronger form of the Riemann hypothesis that is equivalent to the statement that the smallest quadratic nonresidue modulo , for any odd prime , is less than , where here is the natural logarithm.
  • Artin's conjecture on primitive roots: This is a closely related conjecture that states that every integer that is not or a perfect square occurs as a primitive root for infinitely many primes.

Proof

Given: An odd prime , is the smallest positive integer that is a quadratic nonresidue modulo .

To prove: .

Proof: Let be the smallest positive integer such that , and . Since is prime, , so . Further, by the multiplicativity of Legendre symbols:

.

Since is a quadratic nonresidue mod , the first term is , so at least one of and is a nonresidue. Since and is the least nonresidue by assumption, must be a quadratic residue, forcing to be a quadratic nonresidue, and hence, . Since , we get . In particular, since , from which we can deduce that .