Bilangan prima Wolstenholme: Perbedaan antara revisi
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Dua bilangan prima Wolstenholme yang diketahui hanyalah 16843 dan 2124679 {{OEIS|A088164}}. Tiada bilangan prima Wolstenholme yang lebih kecil daripada 10<sup>9</sup>.<ref>{{MathWorld|urlname=WolstenholmePrime|title=Wolstenholme prime|mode=cs2}}</ref>
==Definisi==
{{unsolved|matematika|Adakah bilangan prima Wolstenholme selain 16843 dan 2124679?}}
Bilangan prima Wolstenholme dapat didefinisikan sebagai bilangan prima <math> p > 7 </math> yang memenuhi [[relasi kekongruenan|kekongruenan]]:<math display="block">{2p-1 \choose p-1} \equiv 1 \pmod{p^4}.</math>
Disini, ekspresi di ruas kiri melambangkan [[koefisien binomial]].<ref>{{citation|url = http://www.johndcook.com/binomial_coefficients.html | title = Binomial coefficients | first = J. D. | last = Cook | access-date = 21 December 2010}}</ref>
Sebagai perbandingan, [[teorema Wolstenholme]] menyatakan bahwa untuk setiap bilangan prima <math> p > 3 </math>, maka berlaku kekongruenan:
<math display="block">{2p-1 \choose p-1} \equiv 1 \pmod{p^3}.</math>
Bilangan prima Wolstenholme didefinisikan sebagai bilangan prima <math> p </math> yang membagi pembilang dari [[bilangan Bernoulli]] <math> B_3 </math>.<ref>{{harvnb|Clarke|Jones|2004|p=553}}; {{harvnb|McIntosh|1995|p=387}}; {{harvnb|Zhao|2008|p=25}}.</ref> Karena itu, bilangan prima Wolstenholme membentuk subhimpunan dari [[bilangan prima tak beraturan]]. Bilangan prima Wolstenholme merupakan bilangan prima <math> p </math> sehingga <math> (p,p - 3) </math> merupakan [[Bilangan prima beraturan#Pasangan tak beraturan|pasangan tak beraturan]].<ref>{{harvnb|Johnson|1975|p=114}}; {{harvnb|Buhler|Crandall|Ernvall|Metsänkylä|1993|p=152}}.</ref>
Bilangan prima Wolstenholme adalah bilangan prima <math> p </math> sehingga
<math display="block">H_{p - 1} \equiv 0 \pmod{p^3}.</math>
Ini berarti, pembilang dari [[bilangan harmonik]] <math>H_{p-1}</math> yang dinyatakan dalam suku terkecil dapat dibagi oleh <math> p^3 </math>.{{sfn|Zhao|2007|p=18}}
== Catatan kaki ==
Baris 22 ⟶ 35:
== Referensi ==
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* {{Citation | last1=Zhao | first1=J. | title=Bernoulli numbers, Wolstenholme's theorem, and p<sup>5</sup> variations of Lucas' theorem | year=2007 | journal=Journal of Number Theory | volume=123 | pages=18–26 | doi=10.1016/j.jnt.2006.05.005 | s2cid=937685 | url=http://home.eckerd.edu/~zhaoj/research/ZhaoJNTBern.pdf| doi-access=free }}{{webarchive|url=https://www.webcitation.org/5uBfrPYe8|date=12 November 2010}}
* {{Citation | last1=Zhao | first1=J. | title=Wolstenholme Type Theorem for Multiple Harmonic Sums | year=2008 | journal=International Journal of Number Theory | volume=4 | issue=1 | pages=73–106 | url=http://home.eckerd.edu/~zhaoj/research/ZhaoIJNT.pdf | doi=10.1142/s1793042108001146}} {{webarchive|url=https://www.webcitation.org/5uXu6BHu7|date=27 November 2010}}
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