Higgsovo praštevilo

Higgsovo praštevilo je praštevilo p za katerega p-1 deli kvadrat produkta manjših Higgsovih praštevil brez ostanka. 13 je na primer Higgsovo praštevilo, ker je kvadrat produkta manjših Higgsovih praštevil:

deljiv brez ostanka z 12:

17 ni Higgsovo praštevilo, saj je ostanek pri deljenju kvadrata produktov manjših Higgsovih praštevil 901800900 s 16 enak 4 (901800900 je kongruentno 4 mod (17 - 1)).

Kvadrate je moč posplošiti na kube, četrte potence, itd. Za dani eksponent a za Higgsovo praštevilo velja:

kjer je φ(x) aritmetična Eulerjeva funkcija φ. Po dogovoru je najmanjše Higgsovo praštevilo 2.

Za kvadrate so prva Higgsova praštevila (OEIS A007459):

2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 101, 107, ...

Ni znano ali obstaja neskončno mnogo Higgsovih praštevil za poljuben eksponent a večji od 1. Razmere za a = 1 se precej razlikujejo, saj obstajajo le štiri takšna praštevila: 2, 3, 7 in 43: 2 · 3 · 7 / (43 - 1) = 1. Zaporedje je enako prvim štirim členom Sylvestrovega zaporedja. Velja sicer: 2 · 3 · 7 · 43 / (1807 - 1) = 1, vendar 1807 = 13 · 139 ni praštevilo. Je pa praštevilo spet šesti člen Sylvestrovega zaporedja, sedmi člen, 10650056950807 = 547 · 607 · 1033 · 31051, in osmi, 113423713055421844361000443 = 29881 · 67003 · 9119521 · 6212157481, pa spet nista praštevili. Prav tako niso praštevila deveti do devetnajsti člen Sylvestrovega zaporedja.

Leta 1993 sta S. Burris in S. Lee odkrila da je približno petina praštevil manjših od milijon Higgsovih praštevil in zaključila da četudi je zaporedje Higgsovih praštevil za kvadrate končno, »izračun z računalnikom ni izvedljiv«.

Praštevila, ki niso Higgsova praštevila

Praštevila, ki niso Higgsova praštevila, je moč za eksponente do 7 bolj zgoščeno prikazati.

Eksponent 75-to Higgsovo praštevilo Praštevila, ki niso Higgsova praštevila, manjša od 75-tega Higgsovega praštevila
2 827 17, 41, 73, 83, 89, 97, 103, 109, 113, 137, 163, 167, 179, 193, 227, 233, 239, 241, 251, 257, 271, 281, 293, 307, 313, 337, 353, 359, 379, 389, 401, 409, 433, 439, 443, 449, 457, 467, 479, 487, 499, 503, 521, 541, 563, 569, 577, 587, 593, 601, 613, 617, 619, 641, 647, 653, 673, 719, 739, 751, 757, 761, 769, 773, 809, 811, 821, 823 (OEIS A126790)
3 521 17, 97, 103, 113, 137, 163, 193, 227, 239, 241, 257, 307, 337, 353, 389, 401, 409, 433, 443, 449, 479, 487
4 419 97, 193, 257, 353, 389
5 397 193, 257
6 389 257
7 389 257

Fermatovo praštevilo ne more biti Higgsovo praštevilo za potence a, če je a manjši od 2n.

Viri

  • S. Burris in S. Lee, »Tarski's high school identities«, Amer. Math. Monthly 100 (1993): 233
  • N. Sloane in S. Plouffe, The Encyclopedia of Integer Sequences, New York: Academic Press (1995): M0660

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