Editing Regular prime (section)

== 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>