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{{Short description|Type of prime number}}
{{distinguish|regular number}}
{{unsolved|mathematics|Are there infinitely many regular primes, and if so, is their relative density <math>e^{-1/2}</math>?}}

In [[number theory]], a '''regular prime''' is a special kind of [[prime number]], defined by [[Ernst Kummer]] in 1850 to prove certain cases of [[Fermat's Last Theorem]]. Regular primes may be defined via the [[divisibility]] of either [[class number (number theory)|class numbers]] or of [[Bernoulli number]]s.

The first few regular odd primes are:
: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, ... {{OEIS|id=A007703}}.

== History and motivation ==
In 1850, Kummer proved that [[Fermat's Last Theorem]] is true for a prime exponent ''p'' if ''p'' is regular. This focused attention on the irregular primes.<ref name="Gardiner1988">{{Citation | last1=Gardiner | first1=A. | title=Four Problems on Prime Power Divisibility | year=1988 | journal=American Mathematical Monthly | volume=95 | issue=10 | pages=926–931 | doi=10.2307/2322386| jstor=2322386 }}</ref> In 1852, [[Angelo Genocchi|Genocchi]] was able to prove that the [[First case of Fermat's last theorem|first case of Fermat's Last Theorem]] is true for an exponent ''p'', if {{nowrap|(''p'', ''p'' − 3)}} is not an irregular pair. Kummer improved this further in 1857 by showing that for the "first case" of Fermat's Last Theorem (see [[Sophie Germain's theorem]]) it is sufficient to establish that either {{nowrap|(''p'', ''p'' − 3)}} or {{nowrap|(''p'', ''p'' − 5)}} fails to be an irregular pair.

({{nowrap|(''p'', 2''k'')}} is an irregular pair when ''p'' is irregular due to a certain condition described below being realized at 2''k''.)

Kummer found the irregular primes less than 165. In 1963, Lehmer reported results up to 10000 and Selfridge and Pollack announced in 1964 to have completed the table of irregular primes up to 25000. Although the two latter tables did not appear in print, Johnson found that {{nowrap|(''p'', ''p'' − 3)}} is in fact an irregular pair for {{nowrap|''p'' {{=}} 16843}} and that this is the first and only time this occurs for {{nowrap|''p'' < 30000}}.<ref>{{Citation | last1=Johnson | first1=W. | title=Irregular Primes and Cyclotomic Invariants | year=1975 | journal=[[Mathematics of Computation]] | volume=29 | issue=129 | pages=113–120 | url=https://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0376606-9/ | doi=10.2307/2005468 | jstor=2005468 | doi-access=free }}</ref> It was found in 1993 that the next time this happens is for {{nowrap|''p'' {{=}} 2124679}}; see [[Wolstenholme prime]].<ref>{{cite journal | last1 = Buhler | first1 = J. | last2 = Crandall | first2 = R. | last3 = Ernvall | first3 = R. | last4 = Metsänkylä | first4 = T. | year = 1993 | title = Irregular primes and cyclotomic invariants to four million | journal = Math. Comp. | volume = 61 | issue = 203 | pages = 151–153 | doi=10.1090/s0025-5718-1993-1197511-5| bibcode = 1993MaCom..61..151B | doi-access = free }}</ref>

== Definition ==

=== Class number criterion ===
An odd prime number ''p'' is defined to be regular if it does not divide the [[class number (number theory)|class number]] of the ''p''th [[cyclotomic field]] '''Q'''(''ζ''<sub>''p''</sub>), where ''ζ''<sub>''p''</sub> is a primitive ''p''th root of unity.

The prime number 2 is often considered regular as well.

The [[class number (number theory)|class number]] of the cyclotomic
field is the number of [[ideal (ring theory)|ideals]] of the [[ring of integers]]
'''Z'''(''ζ''<sub>''p''</sub>) up to equivalence. Two ideals ''I'', ''J'' are considered equivalent if there is a nonzero ''u'' in '''Q'''(''ζ''<sub>''p''</sub>) so that {{nowrap|1=''I'' = ''uJ''}}. The first few of these class numbers are listed in {{oeis|id=A000927}}.

=== Kummer's criterion ===
[[Ernst Kummer]] {{harv|Kummer|1850}} showed that an equivalent [[Bernoulli number#The Kummer theorems|criterion]] for regularity is that ''p'' does not divide the numerator of any of the [[Bernoulli number]]s ''B''<sub>''k''</sub> for {{nowrap|''k'' {{=}} 2, 4, 6, ..., ''p'' &minus; 3}}.

Kummer's proof that this is equivalent to the class number definition is strengthened by the [[Herbrand–Ribet theorem]], which states certain consequences of ''p'' dividing the numerator of one of these Bernoulli numbers.

== Siegel's conjecture ==
It has been [[conjecture]]d that there are [[Infinite set|infinitely]] many regular primes. More precisely {{harvs|first=Carl Ludwig|last=Siegel|authorlink=Carl Ludwig Siegel|year=1964|txt}} conjectured  that ''[[e (mathematical constant)|e]]''<sup>−1/2</sup>, or about 60.65%, of all prime numbers are regular, in the [[Asymptotic analysis|asymptotic]] sense of [[natural density]]. 

Taking Kummer's criterion, the chance that one numerator of the Bernoulli numbers <math>B_k</math>, <math>k=2,\dots,p-3</math>, is not divisible by the prime <math>p</math> is

:<math>\dfrac{p-1}{p}</math>

so that the chance that none of the numerators of these Bernoulli numbers are divisible by the prime <math>p</math> is

:<math>\left(\dfrac{p-1}{p}\right)^{\dfrac{p-3}{2}}=\left(1-\dfrac{1}{p}\right)^{\dfrac{p-3}{2}}=\left(1-\dfrac{1}{p}\right)^{-3/2}\cdot\left\lbrace\left(1-\dfrac{1}{p}\right)^{p}\right\rbrace^{1/2}</math>.

By [[E_(mathematical_constant)]], we have

:<math>\lim_{p\to\infty}\left(1-\dfrac{1}{p}\right)^{p}=\dfrac{1}{e}</math>

so that we obtain the probability

:<math>\lim_{p\to\infty}\left(1-\dfrac{1}{p}\right)^{-3/2}\cdot\left\lbrace\left(1-\dfrac{1}{p}\right)^{p}\right\rbrace^{1/2}=e^{-1/2}\approx0.606531</math>.

It follows that about <math>60.6531\%</math> of the primes are regular by chance. Hart et al.<ref>[https://arxiv.org/abs/1605.02398 Irregular primes to two billion, William Hart, David Harvey and Wilson Ong,9 May 2016, arXiv:1605.02398v1]</ref> indicate that <math>60.6590\%</math> of the primes less than <math>2^{31}=2,147,483,648</math> are regular.

== Irregular primes ==
An odd prime that is not regular is an '''irregular prime''' (or Bernoulli irregular or B-irregular to distinguish from other types of irregularity discussed below). The first few irregular primes are:
: 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, ... {{OEIS|id=A000928}}

=== Infinitude ===
[[Kaj Løchte Jensen|K. L. Jensen]] (a student of [[Niels Nielsen (mathematician)|Nielsen]]<ref>[http://tau.ac.il/~corry/publications/articles/pdf/Computers%20and%20FLT.pdf Leo Corry: Number Crunching vs. Number Theory: Computers and FLT, from Kummer to SWAC (1850–1960), and beyond]</ref>) proved in 1915 that there are infinitely many irregular primes of the form {{nowrap|4''n'' + 3}}.<ref>{{cite journal | last = Jensen | first = K. L. | title = Om talteoretiske Egenskaber ved de Bernoulliske Tal | jstor=24532219 | journal = NYT Tidsskr. Mat. | volume = B 26 | pages = 73–83 | year = 1915}}</ref>
In 1954 [[Leonard Carlitz|Carlitz]] gave a simple proof of the weaker result that there are in general infinitely many irregular primes.<ref>{{cite journal | last = Carlitz | first = L. | title = Note on irregular primes | journal = Proceedings of the American Mathematical Society | volume = 5 | issue = 2 | pages = 329–331 | publisher = [[American Mathematical Society|AMS]] | year = 1954 | url = https://www.ams.org/journals/proc/1954-005-02/S0002-9939-1954-0061124-6/S0002-9939-1954-0061124-6.pdf  | issn = 1088-6826 | doi = 10.1090/S0002-9939-1954-0061124-6 | mr = 61124| doi-access = free }}</ref>

Metsänkylä proved in 1971 that for any integer {{nowrap|''T'' > 6}}, there are infinitely many irregular primes not of the form {{nowrap|''mT'' + 1}} or {{nowrap|''mT'' − 1}},<ref>{{cite journal |author=Tauno Metsänkylä |title=Note on the distribution of irregular primes |journal=Ann. Acad. Sci. Fenn. Ser. A I |volume=492 |year=1971 |mr=0274403}}</ref> and later generalized this.<ref>{{cite journal |author=Tauno Metsänkylä |title=Distribution of irregular prime numbers |journal=Journal für die reine und angewandte Mathematik |volume=1976 |issue=282 |doi=10.1515/crll.1976.282.126 |url=http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002191873 |year=1976|pages=126–130 |s2cid=201061944 }}</ref>

=== Irregular pairs ===
If ''p'' is an irregular prime and ''p'' divides the numerator of the Bernoulli number ''B''<sub>2''k''</sub> for {{nowrap|0 < 2''k'' < ''p'' − 1}}, then {{nowrap|(''p'', 2''k'')}} is called an '''irregular pair'''.  In other words, an irregular pair is a bookkeeping device to record, for an irregular prime ''p'', the particular indices of the Bernoulli numbers at which regularity fails. The first few irregular pairs (when ordered by ''k'') are:
: (691, 12), (3617, 16), (43867, 18), (283, 20), (617, 20), (131, 22), (593, 22), (103, 24), (2294797, 24), (657931, 26), (9349, 28), (362903, 28), ... {{OEIS|id=A189683}}.

The smallest even ''k'' such that ''n''th irregular prime divides ''B<sub>k</sub> are
: 32, 44, 58, 68, 24, 22, 130, 62, 84, 164, 100, 84, 20, 156, 88, 292, 280, 186, 100, 200, 382, 126, 240, 366, 196, 130, 94, 292, 400, 86, 270, 222, 52, 90, 22, ... {{OEIS|id=A035112}}

For a given prime ''p'', the number of such pairs is called the '''index of irregularity''' of ''p''.<ref name=Nark475>{{citation | last=Narkiewicz | first=Władysław | title=Elementary and analytic theory of algebraic numbers | edition=2nd, substantially revised and extended | publisher=[[Springer-Verlag]]; [[Polish Scientific Publishers PWN|PWN-Polish Scientific Publishers]] | year=1990 | isbn=3-540-51250-0 | zbl=0717.11045 | page=[https://archive.org/details/elementaryanalyt0000nark/page/475 475] | url=https://archive.org/details/elementaryanalyt0000nark/page/475 }}</ref>  Hence, a prime is regular if and only if its index of irregularity is zero. Similarly, a prime is irregular if and only if its index of irregularity is positive.

It was discovered that {{nowrap|(''p'', ''p'' − 3)}} is in fact an irregular pair for {{nowrap|''p'' {{=}} 16843}}, as well as for {{nowrap|''p'' {{=}} 2124679}}. There are no more occurrences for {{nowrap|''p'' < 10<sup>9</sup>}}.

=== Irregular index ===
An odd prime ''p'' has '''irregular index''' ''n'' [[if and only if]] there are ''n'' values of ''k'' for which ''p'' divides ''B''<sub>2''k''</sub> and these ''k''s are less than {{nowrap|(''p'' − 1)/2}}. The first irregular prime with irregular index greater than 1 is [[157 (number)|157]], which divides ''B''<sub>62</sub> and ''B''<sub>110</sub>, so it has an irregular index 2. Clearly, the irregular index of a regular prime is 0.

The irregular index of the ''n''th prime is
:0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 2, 0, ... (Start with ''n'' = 2, or the prime = 3) {{OEIS|id=A091888}}

The irregular index of the ''n''th irregular prime is
:1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 3, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, ... {{OEIS|id=A091887}}

The primes having irregular index 1 are
: 37, 59, 67, 101, 103, 131, 149, 233, 257, 263, 271, 283, 293, 307, 311, 347, 389, 401, 409, 421, 433, 461, 463, 523, 541, 557, 577, 593, 607, 613, 619, 653, 659, 677, 683, 727, 751, 757, 761, 773, 797, 811, 821, 827, 839, 877, 881, 887, 953, 971, ... {{OEIS|id=A073276}}

The primes having irregular index 2 are
: 157, 353, 379, 467, 547, 587, 631, 673, 691, 809, 929, 1291, 1297, 1307, 1663, 1669, 1733, 1789, 1933, 1997, 2003, 2087, 2273, 2309, 2371, 2383, 2423, 2441, 2591, 2671, 2789, 2909, 2957, ... {{OEIS|id=A073277}}

The primes having irregular index 3 are
: 491, 617, 647, 1151, 1217, 1811, 1847, 2939, 3833, 4003, 4657, 4951, 6763, 7687, 8831, 9011, 10463, 10589, 12073, 13217, 14533, 14737, 14957, 15287, 15787, 15823, 16007, 17681, 17863, 18713, 18869, ... {{OEIS|id=A060975}}

The least primes having irregular index ''n'' are
: 2, 3, 37, 157, 491, 12613, 78233, 527377, 3238481, ... {{OEIS|id=A061576}} (This sequence defines "the irregular index of 2" as −1, and also starts at {{nowrap|1=''n'' = −1}}.)

== Generalizations==

=== Euler irregular primes ===
Similarly, we can define an '''Euler irregular prime''' (or E-irregular) as a prime ''p'' that divides at least one [[Euler number]] ''E''<sub>2''n''</sub> with {{nowrap|0 < 2''n'' ≤ ''p'' − 3}}. The first few Euler irregular primes are
:19, 31, 43, 47, 61, 67, 71, 79, 101, 137, 139, 149, 193, 223, 241, 251, 263, 277, 307, 311, 349, 353, 359, 373, 379, 419, 433, 461, 463, 491, 509, 541, 563, 571, 577, 587, ... {{OEIS|id=A120337}}

The Euler irregular pairs are
: (61, 6), (277, 8), (19, 10), (2659, 10), (43, 12), (967, 12), (47, 14), (4241723, 14), (228135437, 16), (79, 18), (349, 18), (84224971, 18), (41737, 20), (354957173, 20), (31, 22), (1567103, 22), (1427513357, 22), (2137, 24), (111691689741601, 24), (67, 26), (61001082228255580483, 26), (71, 28), (30211, 28), (2717447, 28), (77980901, 28), ...

Vandiver proved in 1940 that [[Fermat's Last Theorem]] ({{nowrap|1=''x''<sup>''p''</sup> + ''y''<sup>''p''</sup> = ''z''<sup>''p''</sup>}}) has no solution for integers ''x'', ''y'', ''z'' with {{nowrap|1=gcd(''xyz'', ''p'') = 1}} if ''p'' is Euler-regular. Gut proved that {{nowrap|1=''x''<sup>2''p''</sup> + ''y''<sup>2''p''</sup> = ''z''<sup>2''p''</sup>}} has no solution if ''p'' has an E-irregularity index less than 5.<ref>{{Cite web|title=The Top Twenty: Euler Irregular primes|url=https://primes.utm.edu/top20/page.php?id=25|access-date=2021-07-21|website=primes.utm.edu}}</ref>

It was proven that there is an infinity of E-irregular primes. A stronger result was obtained: there is an infinity of E-irregular primes [[Modular arithmetic|congruent]] to 1 modulo 8. As in the case of Kummer's B-regular primes, there is as yet no proof that there are infinitely many E-regular primes, though this seems likely to be true.

=== Strong irregular primes ===
A prime ''p'' is called '''strong irregular''' if it is both B-irregular and E-irregular (the indexes of Bernoulli and Euler numbers that are divisible by ''p'' can be either the same or different). The first few strong irregular primes are
: 67, 101, 149, 263, 307, 311, 353, 379, 433, 461, 463, 491, 541, 577, 587, 619, 677, 691, 751, 761, 773, 811, 821, 877, 887, 929, 971, 1151, 1229, 1279, 1283, 1291, 1307, 1319, 1381, 1409, 1429, 1439, ... {{OEIS|A128197}}

To prove the [[Fermat's Last Theorem]] for a strong irregular prime ''p'' is more difficult (since [[Ernst Kummer|Kummer]] proved the first case of Fermat's Last Theorem for B-regular primes, [[Harry Vandiver|Vandiver]] proved the first case of Fermat's Last Theorem for E-regular primes), the most difficult is that ''p'' is not only a strong irregular prime, but {{nowrap|2''p'' + 1}}, {{nowrap|4''p'' + 1}}, {{nowrap|8''p'' + 1}}, {{nowrap|10''p'' + 1}}, {{nowrap|14''p'' + 1}}, and {{nowrap|16''p'' + 1}} are also all composite ([[Adrien-Marie Legendre|Legendre]] proved the first case of Fermat's Last Theorem for primes ''p'' such that at least one of {{nowrap|2''p'' + 1}}, {{nowrap|4''p'' + 1}}, {{nowrap|8''p'' + 1}}, {{nowrap|10''p'' + 1}}, {{nowrap|14''p'' + 1}}, and {{nowrap|16''p'' + 1}} is prime), the first few such ''p'' are
: 263, 311, 379, 461, 463, 541, 751, 773, 887, 971, 1283, ...

=== Weak irregular primes ===
A prime ''p'' is '''weak irregular''' if it is either B-irregular or E-irregular (or both). The first few weak irregular primes are
: 19, 31, 37, 43, 47, 59, 61, 67, 71, 79, 101, 103, 131, 137, 139, 149, 157, 193, 223, 233, 241, 251, 257, 263, 271, 277, 283, 293, 307, 311, 347, 349, 353, 373, 379, 389, 401, 409, 419, 421, 433, 461, 463, 491, 509, 523, 541, 547, 557, 563, 571, 577, 587, 593, ... {{OEIS|id=A250216}}

Like the Bernoulli irregularity, the weak regularity relates to the divisibility of class numbers of [[cyclotomic field]]s. In fact, a prime ''p'' is weak irregular if and only if ''p'' divides the class number of the 4''p''th cyclotomic field '''Q'''(''ζ''<sub>4''p''</sub>).

==== Weak irregular pairs ====
In this section, "''a<sub>n</sub>''" means the numerator of the ''n''th Bernoulli number if ''n'' is even, "''a<sub>n</sub>''" means the {{nowrap|(''n'' − 1)}}th Euler number if ''n'' is odd {{OEIS|id=A246006}}.

Since for every odd prime ''p'', ''p'' divides ''a<sub>p</sub>'' if and only if ''p'' is congruent to 1 mod 4, and since ''p'' divides the denominator of {{nowrap|(''p'' − 1)}}th Bernoulli number for every odd prime ''p'', so for any odd prime ''p'', ''p'' cannot divide ''a''<sub>''p''−1</sub>. Besides, if and only if an odd prime ''p'' divides ''a<sub>n</sub>'' (and 2''p'' does not divide ''n''), then ''p'' also divides ''a''<sub>''n''+''k''(''p''−1)</sub> (if 2''p'' divides ''n'', then the sentence should be changed to "''p'' also divides ''a''<sub>''n''+2''kp''</sub>". In fact, if 2''p'' divides ''n'' and {{nowrap|''p''(''p'' − 1)}} does not divide ''n'', then ''p'' divides ''a''<sub>''n''</sub>.) for every integer ''k'' (a condition is {{nowrap|''n'' + ''k''(''p'' − 1)}} must be >&nbsp;1). For example, since 19 divides ''a''<sub>11</sub> and {{nowrap|1=2 × 19 = 38}} does not divide 11, so 19 divides ''a''<sub>18''k''+11</sub> for all ''k''. Thus, the definition of irregular pair {{nowrap|(''p'', ''n'')}}, ''n'' should be at most {{nowrap|''p'' − 2}}.

The following table shows all irregular pairs with odd prime {{nowrap|''p'' ≤ 661}}:

{|class="wikitable"
|''p''
|integers<br>0 ≤ ''n'' ≤ ''p'' − 2<br>such that ''p'' divides ''a<sub>n</sub>''
|''p''
|integers<br>0 ≤ ''n'' ≤ ''p'' − 2<br>such that ''p'' divides ''a<sub>n</sub>''
|''p''
|integers<br>0 ≤ ''n'' ≤ ''p'' − 2<br>such that ''p'' divides ''a<sub>n</sub>''
|''p''
|integers<br>0 ≤ ''n'' ≤ ''p'' − 2<br>such that ''p'' divides ''a<sub>n</sub>''
|''p''
|integers<br>0 ≤ ''n'' ≤ ''p'' − 2<br>such that ''p'' divides ''a<sub>n</sub>''
|''p''
|integers<br>0 ≤ ''n'' ≤ ''p'' − 2<br>such that ''p'' divides ''a<sub>n</sub>''
|-
|3
|
|79
|19
|181
|
|293
|156
|421
|240
|557
|222
|-
|5
|
|83
|
|191
|
|307
|88, 91, 137
|431
|
|563
|175, 261
|-
|7
|
|89
|
|193
|75
|311
|87, 193, 292
|433
|215, 366
|569
|
|-
|11
|
|97
|
|197
|
|313
|
|439
|
|571
|389
|-
|13
|
|101
|63, 68
|199
|
|317
|
|443
|
|577
|52, 209, 427
|-
|17
|
|103
|24
|211
|
|331
|
|449
|
|587
|45, 90, 92
|-
|19
|11
|107
|
|223
|133
|337
|
|457
|
|593
|22
|-
|23
|
|109
|
|227
|
|347
|280
|461
|196, 427
|599
|
|-
|29
|
|113
|
|229
|
|349
|19, 257
|463
|130, 229
|601
|
|-
|31
|23
|127
|
|233
|84
|353
|71, 186, 300
|467
|94, 194
|607
|592
|-
|37
|32
|131
|22
|239
|
|359
|125
|479
|
|613
|522
|-
|41
|
|137
|43
|241
|211, 239
|367
|
|487
|
|617
|20, 174, 338
|-
|43
|13
|139
|129
|251
|127
|373
|163
|491
|292, 336, 338, 429
|619
|371, 428, 543
|-
|47
|15
|149
|130, 147
|257
|164
|379
|100, 174, 317
|499
|
|631
|80, 226
|-
|53
|
|151
|
|263
|100, 213
|383
|
|503
|
|641
|
|-
|59
|44
|157
|62, 110
|269
|
|389
|200
|509
|141
|643
|
|-
|61
|7
|163
|
|271
|84
|397
|
|521
|
|647
|236, 242, 554
|-
|67
|27, 58
|167
|
|277
|9
|401
|382
|523
|400
|653
|48
|-
|71
|29
|173
|
|281
|
|409
|126
|541
|86, 465
|659
|224
|-
|73
|
|179
|
|283
|20
|419
|159
|547
|270, 486
|661
|
|}

The only primes below 1000 with weak irregular index 3 are 307, 311, 353, 379, 577, 587, 617, 619, 647, 691, 751, and 929. Besides, 491 is the only prime below 1000 with weak irregular index 4, and all other odd primes below 1000 with weak irregular index 0, 1, or 2. ('''Weak irregular index''' is defined as "number of integers {{nowrap|0 ≤ ''n'' ≤ ''p'' − 2}} such that ''p'' divides ''a<sub>n</sub>''.)

The following table shows all irregular pairs with ''n'' ≤ 63. (To get these irregular pairs, we only need to factorize ''a<sub>n</sub>''. For example, {{nowrap|1=''a''<sub>34</sub> = 17 × 151628697551}}, but {{nowrap|17 < 34 + 2}}, so the only irregular pair with {{nowrap|1=''n'' = 34}} is {{nowrap|(151628697551, 34)}}) (for more information (even ''n''s up to 300 and odd ''n''s up to 201), see <ref>{{Cite web|title=Bernoulli and Euler numbers|url=https://homes.cerias.purdue.edu/~ssw/bernoulli/index.html|access-date=2021-07-21|website=homes.cerias.purdue.edu}}</ref>).

{|class="wikitable"
|''n''
|primes ''p'' ≥ ''n'' + 2 such that ''p'' divides ''a<sub>n</sub>''
|''n''
|primes ''p'' ≥ ''n'' + 2 such that ''p'' divides ''a<sub>n</sub>''
|-
|0
|
|32
|37, 683, 305065927
|-
|1
|
|33
|930157, 42737921, 52536026741617
|-
|2
|
|34
|151628697551
|-
|3
|
|35
|4153, 8429689, 2305820097576334676593
|-
|4
|
|36
|26315271553053477373
|-
|5
|
|37
|9257, 73026287, 25355088490684770871
|-
|6
|
|38
|154210205991661
|-
|7
|61
|39
|23489580527043108252017828576198947741
|-
|8
|
|40
|137616929, 1897170067619
|-
|9
|277
|41
|763601, 52778129, 359513962188687126618793
|-
|10
|
|42
|1520097643918070802691
|-
|11
|19, 2659
|43
|137, 5563, 13599529127564174819549339030619651971
|-
|12
|691
|44
|59, 8089, 2947939, 1798482437
|-
|13
|43, 967
|45
|587, 32027, 9728167327, 36408069989737, 238716161191111
|-
|14
|
|46
|383799511, 67568238839737
|-
|15
|47, 4241723
|47
|285528427091, 1229030085617829967076190070873124909
|-
|16
|3617
|48
|653, 56039, 153289748932447906241
|-
|17
|228135437
|49
|5516994249383296071214195242422482492286460673697
|-
|18
|43867
|50
|417202699, 47464429777438199
|-
|19
|79, 349, 87224971
|51
|5639, 1508047, 10546435076057211497, 67494515552598479622918721
|-
|20
|283, 617
|52
|577, 58741, 401029177, 4534045619429
|-
|21
|41737, 354957173
|53
|1601, 2144617, 537569557577904730817, 429083282746263743638619
|-
|22
|131, 593
|54
|39409, 660183281, 1120412849144121779
|-
|23
|31, 1567103, 1427513357
|55
|2749, 3886651, 78383747632327, 209560784826737564385795230911608079
|-
|24
|103, 2294797
|56
|113161, 163979, 19088082706840550550313
|-
|25
|2137, 111691689741601
|57
|5303, 7256152441, 52327916441, 2551319957161, 12646529075062293075738167
|-
|26
|657931
|58
|67, 186707, 6235242049, 37349583369104129
|-
|27
|67, 61001082228255580483
|59
|1459879476771247347961031445001033, 8645932388694028255845384768828577
|-
|28
|9349, 362903
|60
|2003, 5549927, 109317926249509865753025015237911
|-
|29
|71, 30211, 2717447, 77980901
|61
|6821509, 14922423647156041, 190924415797997235233811858285255904935247
|-
|30
|1721, 1001259881
|62
|157, 266689, 329447317, 28765594733083851481
|-
|31
|15669721, 28178159218598921101
|63
|101, 6863, 418739, 1042901, 91696392173931715546458327937225591842756597414460291393
|}

The following table shows irregular pairs {{nowrap|(''p'', ''p'' − ''n'')}} ({{nowrap|''n'' ≥ 2}}), it is a conjecture that there are infinitely many irregular pairs {{nowrap|(''p'', ''p'' − ''n'')}} for every natural number {{nowrap|''n'' ≥ 2}}, but only few were found for fixed ''n''. For some values of ''n'', even there is no known such prime ''p''.

{|class="wikitable"
|''n''
|primes ''p'' such that ''p'' divides ''a''<sub>''p''−''n''</sub> (these ''p'' are checked up to 20000)
|[[OEIS]] sequence
|-
|2
|149, 241, 2946901, 16467631, 17613227, 327784727, 426369739, 1062232319, ...
|{{OEIS link|id=A198245}}
|-
|3
|16843, 2124679, ...
|{{OEIS link|id=A088164}}
|-
|4
|...
|
|-
|5
|37, ...
|
|-
|6
|...
|
|-
|7
|...
|
|-
|8
|19, 31, 3701, ...
|
|-
|9
|67, 877, ...
|{{OEIS link|id=A212557}}
|-
|10
|139, ...
|
|-
|11
|9311, ...
|
|-
|12
|...
|
|-
|13
|...
|
|-
|14
|...
|
|-
|15
|59, 607, ...
|
|-
|16
|1427, 6473, ...
|
|-
|17
|2591, ...
|
|-
|18
|...
|
|-
|19
|149, 311, 401, 10133, ...
|
|-
|20
|9643, ...
|
|-
|21
|8369, ...
|
|-
|22
|...
|
|-
|23
|...
|
|-
|24
|17011, ...
|
|-
|25
|...
|
|-
|26
|...
|
|-
|27
|...
|
|-
|28
|...
|
|-
|29
|4219, 9133, ...
|
|-
|30
|43, 241, ...
|
|-
|31
|3323, ...
|
|-
|32
|47, ...
|
|-
|33
|101, 2267, ...
|
|-
|34
|461, ...
|
|-
|35
|...
|
|-
|36
|1663, ...
|
|-
|37
|...
|
|-
|38
|101, 5147, ...
|
|-
|39
|3181, 3529, ...
|
|-
|40
|67, 751, 16007, ...
|
|-
|41
|773, ...
|
|}

== See also ==
* [[Wolstenholme prime]]

== References ==
{{reflist}}

== Further reading ==
{{refbegin}}
* {{citation|first=E. E.|last=Kummer|author-link=Ernst Kummer| title=Allgemeiner Beweis des Fermat'schen Satzes, dass die Gleichung ''x''<sup>''λ''</sup> + ''y''<sup>''λ''</sup> = ''z''<sup>''λ''</sup> durch ganze Zahlen unlösbar ist, für alle diejenigen Potenz-Exponenten ''λ'', welche ungerade Primzahlen sind und in den Zählern der ersten (''λ''−3)/2 Bernoulli'schen Zahlen als Factoren nicht vorkommen |journal=J. Reine Angew. Math. |volume=40 |year=1850 |pages=131–138 |url=http://www.digizeitschriften.de/resolveppn/GDZPPN002146738}}
* {{citation
 | last = Siegel | first = Carl Ludwig | author-link = Carl Ludwig Siegel
 | journal = Nachrichten der Akademie der Wissenschaften in Göttingen
 | mr = 0163899
 | pages = 51–57
 | title = Zu zwei Bemerkungen Kummers
 | volume = 1964
 | year = 1964
 }}
* {{Citation | last1=Iwasawa | first1=K. | last2=Sims | first2=C. C. | title=Computation of invariants in the theory of cyclotomic fields | year=1966 | journal=Journal of the Mathematical Society of Japan | volume=18 | issue=1 | pages=86–96 | url=https://projecteuclid.org/euclid.jmsj/1260541355 | doi=10.2969/jmsj/01810086 | doi-access=free }} 
* {{Citation | last1=Wagstaff, Jr. | first1=S. S. | title=The Irregular Primes to 125000 | year=1978 | journal=[[Mathematics of Computation]] | volume=32 | issue=142 | pages=583–591 | url=https://www.ams.org/journals/mcom/1978-32-142/S0025-5718-1978-0491465-4/| doi=10.2307/2006167 | jstor=2006167 }} 
* {{Citation | last1=Granville | first1=A. | last2=Monagan | first2=M. B. | title=The First Case of Fermat's Last Theorem is True for All Prime Exponents up to 714,591,416,091,389 | year=1988 | journal=Transactions of the American Mathematical Society | volume=306 | issue=1 | pages=329–359 | doi=10.1090/S0002-9947-1988-0927694-5 | mr = 0927694| doi-access=free }}
* {{Citation | last1=Gardiner | first1=A. | title=Four Problems on Prime Power Divisibility | year=1988 | journal=American Mathematical Monthly | volume=95 | issue=10 | pages=926–931 | doi=10.2307/2322386| jstor=2322386 }}
* {{Citation | last1=Ernvall | first1=R. | last2=Metsänkylä | first2=T. | title=Cyclotomic Invariants for Primes Between 125000 and 150000 | year=1991 | journal=[[Mathematics of Computation]] | volume=56 | issue=194 | pages=851–858 | url=https://www.ams.org/journals/mcom/1991-56-194/S0025-5718-1991-1068819-7/ | doi=10.2307/2008413 | jstor=2008413 }} 
* {{Citation | last1=Ernvall | first1=R. | last2=Metsänkylä | first2=T. | title=Cyclotomic Invariants for Primes to One Million | year=1992 | journal=Mathematics of Computation | volume=59 | issue=199 | pages=249–250 | url=https://www.ams.org/journals/mcom/1992-59-199/S0025-5718-1992-1134727-7/S0025-5718-1992-1134727-7.pdf | doi=10.2307/2152994| jstor=2152994 | doi-access=free }}
* {{Citation | last1=Buhler | first1=J. P. | last2=Crandall | first2=R. E. | last3=Sompolski | first3=R. W. | title=Irregular Primes to One Million | year=1992 | journal=[[Mathematics of Computation]] | volume=59 | issue=200 | pages=717–722 | url=https://www.ams.org/journals/mcom/1992-59-200/S0025-5718-1992-1134717-4/ | doi=10.2307/2153086  | jstor=2153086 | doi-access=free }} 
* {{Citation | last1 = Boyd | first1 = D. W.| title = A ''p''-adic Study of the Partial Sums of the Harmonic Series | url = http://projecteuclid.org/euclid.em/1048515811| doi = 10.1080/10586458.1994.10504298 | journal = [[Experimental Mathematics (journal)|Experimental Mathematics]]| volume = 3 | issue = 4 | pages = 287–302 | year = 1994| zbl = 0838.11015}}
* {{Citation | last1=Shokrollahi | first1=M. A. | title=Computation of Irregular Primes up to Eight Million (Preliminary Report) | year=1996 | series=ICSI Technical Report | volume=TR-96-002 | url = http://www.icsi.berkeley.edu/ftp/global/global/pub/techreports/1996/tr-96-002.ps.gz }}
* {{Citation | last1=Buhler | first1=J. | last2=Crandall | first2=R. | last3=Ernvall | first3=R. | last4=Metsänkylä | first4=T. | last5=Shokrollahi | first5=M.A. | title=Irregular Primes and Cyclotomic Invariants to 12 Million | year=2001 | journal=Journal of Symbolic Computation | volume=31 | issue=1–2 | pages=89–96 | doi=10.1006/jsco.1999.1011| doi-access=free }}
* {{Citation |author=Richard K. Guy |author-link=Richard K. Guy |title=Unsolved Problems in Number Theory |edition=3rd |publisher=[[Springer Verlag]] |year=2004 |isbn=0-387-20860-7 |chapter=Section D2. The Fermat Problem}}
* {{Citation | last=Villegas | first=F. R. | title=Experimental Number Theory | publisher=Oxford University Press | year=2007 | location=New York | pages=166–167 | url=https://books.google.com/books?id=xXNFmoEaD9QC&pg=PA166 | isbn=978-0-19-852822-7}}
{{refend}}

== External links ==
* {{mathworld|urlname=IrregularPrime|title=Irregular prime}}
* Chris Caldwell, [http://primes.utm.edu/glossary/page.php?sort=Regular The Prime Glossary: regular prime] at The [[Prime Pages]].
* Keith Conrad, [http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/fltreg.pdf Fermat's last theorem for regular primes].
* [http://primes.utm.edu/top20/page.php?id=26 Bernoulli irregular prime]
* [http://primes.utm.edu/top20/page.php?id=25 Euler irregular prime]
* [http://www.luschny.de/math/primes/irregular.html Bernoulli and Euler irregular primes].
* [http://homes.cerias.purdue.edu/~ssw/bernoulli/index.html Factorization of Bernoulli and Euler numbers]
* [http://homes.cerias.purdue.edu/~ssw/bernoulli/full.pdf Factorization of Bernoulli and Euler numbers]

{{Prime number classes}}

[[Category:Algebraic number theory]]
[[Category:Cyclotomic fields]]
[[Category:Classes of prime numbers]]
[[Category:Unsolved problems in number theory]]