examples of regular primes

Examples:

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 On-Line Encyclopedia of Integer Sequences, sequence A000928)

2. The following are the first few class numbers of the cyclotomic fields $\Rats(\zeta_p)$ , 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 $\cite{wash}$ .

   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.

Bibliography

1

L. C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York.