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{{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, ... {{OEIS|id=A007703}}.

==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 ''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 isomorphism. Two ideals ''I,J'' are considered isomorphic if there is a nonzero ''u'' in '''Q'''(ζ<sub>''p''</sub>) so that ''I=uJ''.

===Kummer's criterion===
[[Ernst Kummer]] {{harv|Kummer|1850}} showed that an equivalent 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 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>&minus;1/2</sup>, or about 60.65%, of all prime numbers are regular, in the [[Asymptotic analysis|asymptotic]] sense of [[natural density]]. Neither conjecture has been proven since their conception.

== Irregular primes ==
An odd prime that is not regular is an '''irregular prime'''. The first few irregular primes are:

: 37, 59, 67, 101, 103, 131, 149, ... {{OEIS|id=A000928}}

===Infinitude===
[[Kaj Løchte Jensen|K. L. Jensen]] (an unknown 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>) has shown in 1915 that there are infinitely many irregular primes of the form 4''n'' + 3.
<ref>{{cite journal | last = Jensen | first = K. L. | title = Om talteoretiske Egenskaber ved de Bernoulliske Tal | 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 | pages = 329–331 | publisher = [[American Mathematical Society|AMS]] | year = 1954 | url = http://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}}</ref>

Metsänkylä proved<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> that for any integer ''T'' > 6, there are infinitely many irregular primes not of the form {{nowrap|''mT'' + 1}} or {{nowrap|''mT'' − 1}}.

===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 book-keeping device to record, for an irregular prime ''p'', the particular indices of the Bernoulli numbers at which regularity fails. The first few irregular pairs are:

: (691, 12), (3617, 16), (43867, 18), (283, 20), (617, 20), (131, 22), (593, 22), (103, 24), ... {{OEIS|id=A189683}}.

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=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}}.  This is the first and only time this occurs for {{nowrap|''p'' < 30000}}.

==History==
In 1850, Kummer proved that [[Fermat's Last Theorem]] is true for a prime exponent ''p'' if ''p'' is regular. This raised attention in 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}}</ref> In 1852, 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.  

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=http://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0376606-9/S0025-5718-1975-0376606-9.pdf}} {{WebCite|url=http://www.webcitation.org/5v79AhVZp}}</ref>

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

==References==
{{reflist}}

==Further reading==
*{{citation|first=E. E.|last=Kummer|authorlink=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}}
* {{cite journal |author=[[Carl Ludwig Siegel]] |title=Zu zwei Bemerkungen Kummers |journal=Nachr. Akad. d. Wiss. Goettingen, Math. Phys. K1. |volume=II |year=1964 |pages=51–62}}
* {{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=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.jmsj/1260541355 |doi=10.2969/jmsj/01810086}} {{WebCite|url=http://www.webcitation.org/5vchHFmRX}}
* {{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=http://www.ams.org/journals/mcom/1978-32-142/S0025-5718-1978-0491465-4/S0025-5718-1978-0491465-4.pdf}} {{WebCite|url=http://www.webcitation.org/5vcghCvCT}}
* {{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 | mr=0002994788 | url=http://www.ams.org/journals/tran/1988-306-01/S0002-9947-1988-0927694-5/S0002-9947-1988-0927694-5.pdf | doi=10.1090/S0002-9947-1988-0927694-5}}[http://www.webcitation.org/5vaRsEal9 archived at WebCite]
* {{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}}
* {{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=http://www.ams.org/journals/mcom/1991-56-194/S0025-5718-1991-1068819-7/S0025-5718-1991-1068819-7.pdf}} {{WebCite|url=http://www.webcitation.org/5usA4OX0j}}
* {{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=http://www.ams.org/journals/mcom/1992-59-199/S0025-5718-1992-1134727-7/S0025-5718-1992-1134727-7.pdf}}
* {{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=http://www.ams.org/journals/mcom/1992-59-200/S0025-5718-1992-1134717-4/S0025-5718-1992-1134717-4.pdf}} {{WebCite|url=http://www.webcitation.org/5uXvYmWfE}}
* {{cite doi|10.1080/10586458.1994.10504298}}
* {{Citation | last1=Shokrollahi | first1=M. A. | title=Computation of Irregular Primes up to Eight Million (Preliminary Report) | year=1996 | journal=ICSI Technical Report | volume=TR-96-002 | id = {{citeseerx|10.1.1.38.4040}}}}{{WebCite|url=http://www.webcitation.org/5v7C62HjA}}
* {{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}}
* {{cite book |author=[[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}}
* {{Cite book | last=Villegas | first=F. R. | title=Experimental Number Theory | publisher=Oxford University Press | year=2007 | location=New York | pages=166–167 | url=http://books.google.com/books?id=xXNFmoEaD9QC&pg=PA166 | isbn=978-0-19-852822-7}}

== External links ==
* 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].

{{Prime number classes}}

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