# Irregular primes to 163 million

(with Joe Buhler)

Math. Comp. **80** (2011), 2435–2444 (DOI).

arXiv preprint (December 2009).

## Abstract

We compute all irregular primes less than 163,577,856. For all of these primes we verify that the Kummer–Vandiver conjecture holds and that the λ-invariant is equal to the index of irregularity.

## Accompanying data

### List of irregular pairs

indices_163577856.bz2

compressed = 46 MB, uncompressed = 120 MB

uncompressed MD5 checksum = `9c06c892d216ef6e6f0470b2aafd712e`

This file contains one line for each prime *p* < 163577856. Each line contains *p* followed by the irregular indices *k* for *p*. For example, the line

32012327 11016400 12596980 15062986 15329230 19631338 27353368 28255946

means that *p* = 32012327 has index of irregularity 7, and that *B*_{k} = 0 mod *p* for *k* = 11016400, 12596980, 15062986, 15329230, 19631338, 27353368, and 28255946. If a prime is regular, it appears on the line by itself.
### Data for Kummer–Vandiver and cyclotomic invariant checks

checks_163577856.bz2

compressed = 102 MB, uncompressed = 240 MB

uncompressed MD5 checksum = `759c4e08fdb008fcd2afc7acdea4ffc5`

This file contains one line for each prime *p* < 163577856. Each line contains *p* followed by a clump of the form *k*:*v*,*s*,*t* for each irregular index. In the notation of the paper, *v*, *s* and *t* are respectively *V*_{p,k}, *S*(*k* − 1)/*p* and *S*(*k* + *p* − 2)/*p*. Fifteen of the lines contain, in addition to the data above, a (cryptic) comment as to why the line is “interesting” surrounded by semicolons. An example line:

32012327 11016400:175939647,15703970,13838375 12596980:303121954,21688583,20692880 15062986:44038344,28632266,24230253 15329230:364060109,3834460,10374401 19631338:422859016,15465672,6561073 27353368:120022702,24891973,10272561 28255946:22874060,10489245,1763271; Irr = 7;

### Huge table of Bernoulli numbers modulo *p*

huge_163577856.bz2

compressed = 1.48 GB, uncompressed = 3.69 GB

uncompressed MD5 checksum = `516195cd08a65fb392607ef7703b8803`

This file contains one line for each prime *p* < 163577856. Each line contains *p* followed by several pairs of values *k* and *B*_{k} mod *p*. A precise description of how the values were selected may be found in the paper. An example line:

99979777 99888888 2 73383532 3 41479086 4 81870206 4 55454956 7 94230300 10 98230158 13 14744054 19 89672078 21 23780504 23 32871428 24 91130850 24 53791686 25 87614762 25 15076356 26 82352294 28 38595002 30 49680576 30 8354754 34 26746248 34 40942308 34 33472372 35 78455276 36 83290008 37 64959064 39 28090076 49 70878806 49 33832840 50 67360738 52 97460472 62 310704 63 90935840 64 36846348 65 39980740 66 46548826 71 66478598 75 85974154 77

This means that *B*_{99888888} = 2 mod 99979777, *B*_{73383532} = 3 mod 99979777, and so on.

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