# examples of regular primes

Examples:

1. 1.

These are all the irregular primes 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 numbers of the cyclotomic fields $\mathbb{Q}(\zeta_{p})$, where $\zeta_{p}$ is a primitive $p$-th root of unity:

$p$ 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 $41241=3\cdot 59\cdot 233$, 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 ExamplesOfRegularPrimes 2013-03-22 14:05:58 2013-03-22 14:05:58 alozano (2414) alozano (2414) 10 alozano (2414) Example msc 11R18 msc 11R29 ClassNumbersAndDiscriminantsTopicsOnClassGroups