examples of regular primes


Examples:

  1. 1.

    These are all the irregular primesMathworldPlanetmath up to 1061:

    37, 59, 67, 101, 103, 131, 149, 157, 233, 257, 263, 271,
    283, 293, 307, 311, 347, 353, 379, 389, 401,
    409, 421, 433, 461, 463, 467, 491, 523, 541,
    547, 557, 577, 587, 593, 607, 613, 617, 619,
    631, 647, 653, 659, 673, 677, 683, 691, 727,
    751, 757, 761, 773, 797, 809, 811, 821, 827,
    839, 877, 881, 887, 929, 953, 971, 1061.

    (for this, see the http://www.research.att.com/ njas/sequences/Seis.htmlOn-Line Encyclopedia of Integer Sequences, http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.c gi?Anum=A000928 sequence A000928)

  2. 2.

    The following are the first few class numbersMathworldPlanetmathPlanetmath of the cyclotomic fieldsMathworldPlanetmath (ζp), where ζp is a primitive p-th root of unityMathworldPlanetmath:

    p Class Number
    3 1
    5 1
    7 1
    11 1
    13 1
    17 1
    19 1
    23 3
    29 8
    31 9
    37 37
    41 121
    43 211
    47 695
    53 4889
    59 41241
    61 76301

    An excellent reference for this is [1].

    Remarks:

    • Notice that 37 divides 37, and 59 divides 41241=359233, thus 37, 59 are irregular primes (see above).

    • The class number of the cyclotomic fields grows very quickly with p. For example, p=19 is the last cyclotomic field of class number 1.

References

  • 1 L. C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York.
Title examples of regular primes
Canonical name ExamplesOfRegularPrimes
Date of creation 2013-03-22 14:05:58
Last modified on 2013-03-22 14:05:58
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 10
Author alozano (2414)
Entry type Example
Classification msc 11R18
Classification msc 11R29
Related topic ClassNumbersAndDiscriminantsTopicsOnClassGroups