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单词 RegularPrime
释义

regular prime


A prime p is regularPlanetmathPlanetmath if the class numberMathworldPlanetmathPlanetmath of the cyclotomic fieldMathworldPlanetmath (ζp) is not divisible by p (where ζp:=e2πi/p denotes a primitive pth root of unityMathworldPlanetmath). An irregular primeMathworldPlanetmath is a prime that is not regular.

Regular primes rose to prominence as a result of Ernst Kummer’s work in the 1850’s on Fermat’s Last TheoremMathworldPlanetmath. Kummer was able to prove Fermat’s Last Theorem in the case where the exponent is a regular prime, a result that prior to Wiles’s recent work was the only demonstration of Fermat’s Last Theorem for a large class of exponents. In the course of this work Kummer also established the following numerical criterion for determining whether a prime is regular:

  • p is regular if and only if none of the numerators of the Bernoulli numbersMathworldPlanetmathPlanetmath B0, B2, B4,,Bp-3 is a multipleMathworldPlanetmath of p.

Based on this criterion it is possible to give a heuristic argument that the regular primes have density e-1/2 in the set of all primes [1]. Despite this, there is no known proof that the set of regular primes is infiniteMathworldPlanetmath, although it is known that there are infinitely many irregular primes.

References

  • 1 Kenneth Ireland & Michael Rosen, AClassical Introduction to Modern Number TheoryMathworldPlanetmathPlanetmath, Springer-Verlag, NewYork, Second Edition, 1990.
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更新时间:2025/5/5 1:01:35