Alkulukupari

Alkulukupariksi eli alkulukukaksosiksi kutsutaan kahta alkulukua, joiden erotus on 2. Kymmenen ensimmäistä alkulukuparia ovat (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103) ja (107, 109).[1]

Alkulukupareja arvellaan olevan äärettömän monta, mutta tätä ei ole todistettu. 14. toukokuuta 2013 Zhang Yitang New Hampshiren yliopistosta julkaisi todistuksen[2], jonka mukaan on olemassa äärettömän monta alkulukua ja , missä [3] Myöhemmin :n arvo on saatu pudotettua lukuun 246.[4]

Kaikki alkulukuparit lukuun ottamatta paria (3, 5) ovat muotoa (6n − 1, 6n + 1), jossa n on luonnollinen luku, jonka täytyy päättyä numeroon 0, 2, 3, 5, 7 tai 8, lukuun ottamatta tapausta n = 1.

Clementin lauseen mukaan[5] (m, m + 2) on alkulukupari, jos ja vain jos

Lisäksi on todistettu seuraava lause:[6]

Olkoon . Tällöin ja muodostavat alkulukuparin, jos ja vain jos on jaollinen :llä muttei :lla.

Sergusovin lauseen mukaan ja ovat alkulukuja jos ja vain jos

, missä sekä funktio Eulerin funktio ja luvun jakajien summan laskeva Sigma funktio.[7]selvennä[8]

Suurin tunnettu alkulukupari

Suurin tunnettu alkulukupari on 14. syyskuuta 2016 löydetty . Molemmissa alkulukuparin luvuissa on 388 342 numeroa.[9]

Alkulukuparien määrä

n:ää pienempien tai yhtä suurten alkulukuparien määrä

Alkulukupareja arvellaan olevan äärettömän monta, mutta tätä ei ole todistettu. Niiden lukumäärä onkin lukuteorian suurimpia ratkaisemattomia ongelmia. Alkulukupareille on kuitenkin olemassa samankaltainen laskufunktio kuin alkuluvuillekin, , joka ilmaisee lukua n pienempien alkulukuparien määrän.

n
8
35
205
1 224
8 169
58 980
440 312
3 424 506
27 412 679
224 376 048
1 870 585 220
15 834 664 872
135 780 321 665

Katso myös

Lähteet

  1. ↑ A014574 OEIS-tietokannassa
  2. ↑ Yitang Zhang: Bounded gaps between primes. Annals of Mathematics (2), 2014, 179. vsk, s. 1121-1174. Artikkelin verkkoversio.
  3. ↑ First proof that infinitely many prime numbers come in pairs nature.com. 14. toukokuuta 2013. Viitattu 14.5.2013.
  4. ↑ Bounded_gaps_between_primes michaelnielsen.org. 19. huhtikuuta 2014. Viitattu 19.4.2014.
  5. ↑ http://www.math.sunysb.edu/~moira/mat331-spr10/papers/1949%20ClementCongruences%20for%20Sets%20of%20Primes.pdf
  6. ↑ M. Chaves, Twin Primes and a Primality Test by Indivisibility, http://arxiv.org/pdf/math/0211034v3
  7. ↑ http://www.math.snu.ac.kr/~mhkim/speech/speech_44.pdf
  8. ↑ Tomasz Bucher, http://tomasz.buchert.pl/files/math-one.pdf, Pro Gradu, Puola, 2011. s. 26-28
  9. ↑ www.primegrid.com
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