Difference between revisions of "Smallest quadratic nonresidue is less than square root plus one"
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==Related facts== | ==Related facts== | ||
| + | ===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
Contents
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
- 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 .
