Proth prime

  (Redirected from Proth number)

A Proth number is a number N of the form where k and n are positive integers, k is odd and . A Proth prime is a Proth number that is prime. They are named after the French mathematician François Proth.[1] The first few Proth primes are

Proth prime
Named afterFrançois Proth
Publication year1878
Author of publicationProth, Francois
No. of known termsOver 1.5 billion below 270
Conjectured no. of termsInfinite
Subsequence ofProth numbers, prime numbers
Formulak × 2n + 1
First terms3, 5, 13, 17, 41, 97, 113
Largest known term10223 × 231172165 + 1 (as of December 2019)
OEIS index
  • A080076
  • Proth primes: primes of the form k*2^m + 1 with odd k < 2^m, m >= 1
3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857 (OEISA080076).

The primality of Proth numbers can be tested more easily than many other numbers of similar magnitude.

DefinitionEdit

A Proth number takes the form   where k and n are positive integers,   is odd and  . A Proth prime is a Proth number that is prime.[1][2]

Without the condition that  , all odd integers larger than 1 would be Proth numbers.[3]

Primality testingEdit

The primality of a Proth number can be tested with Proth's theorem, which states that a Proth number   is prime if and only if there exists an integer   for which

 [2][4]

This theorem can be used as a probabilistic test of primality, by checking for many random choices of   whether   If this fails to hold for several random  , then it is very likely that the number   is composite.[citation needed] This test is a Las Vegas algorithm: it never returns a false positive but can return a false negative; in other words, it never reports a composite number as "probably prime" but can report a prime number as "possibly composite".

In 2008, Sze created a deterministic algorithm that runs in at most   time, where Õ is the soft-O notation. For typical searches for Proth primes, usually   is either fixed (e.g. 321 Prime Search or Sierpinski Problem) or of order   (e.g. Cullen prime search). In these cases algorithm runs in at most  , or   time for all  . There is also an algorithm that runs in   time.[1][5]

Large primesEdit

As of 2019, the largest known Proth prime is  . It is 9,383,761 digits long.[6] It was found by Szabolcs Peter in the PrimeGrid distributed computing project which announced it on 6 November 2016.[7] It is also the largest known non-Mersenne prime.[8]

The project Seventeen or Bust, searching for Proth primes with a certain   to prove that 78557 is the smallest Sierpinski number (Sierpinski problem), has found 11 large Proth primes by 2007, of which 5 are megaprimes. Similar resolutions to the prime Sierpiński problem and extended Sierpiński problem have yielded several more numbers.

As of December 2019, PrimeGrid is the leading computing project for searching for Proth primes. Its main projects include:

  • general Proth prime search
  • 321 Prime Search (searching for primes of the form  , also called Thabit primes of the second kind)
  • 27121 Prime Search (searching for primes of the form   and  )
  • Cullen prime search (searching for primes of the form  )
  • Sierpinski problem (and their prime and extended generalizations) - searching for primes of the form   where k is in this list:

k ∈ {21181, 22699, 24737, 55459, 67607, 79309, 79817, 91549, 99739, 131179, 152267, 156511, 163187, 200749, 202705, 209611, 222113, 225931, 227723, 229673, 237019, 238411}

As of December 2019, the largest Proth primes discovered are:[9]

rank prime digits when Cullen Prime? Discoverer (Project) References
1 10223 · 231172165 + 1 9383761 31 Oct 2016 Szabolcs Péter (Sierpinski Problem) [10]
2 168451 · 219375200 + 1 5832522 17 Sep 2017 Ben Maloney (Prime Sierpinski Problem) [11]
3 19249 · 213018586 + 1 3918990 26 Mar 2007 Konstantin Agafonov (Seventeen or Bust) [10]
4 193997 · 211452891 + 1 3447670 3 Apr 2018 Tom Greer (Extended Sierpinski Problem) [12]
5 3 · 210829346 + 1 3259959 14 Jan 2014 Sai Yik Tang (321 Prime Search) [13]
6 27653 · 29167433 + 1 2759677 8 Jun 2005 Derek Gordon (Seventeen or Bust) [10]
7 90527 · 29162167 + 1 2758093 30 Jun 2010 Unknown (Prime Sierpinski Problem) [14][15]
8 28433 · 27830457 + 1 2357207 30 Dec 2004 Team Prime Rib (Seventeen or Bust) [10]
9 161041 · 27107964 + 1 2139716 6 Jan 2015 Martin Vanc (Extended Sierpinski Problem) [16]
10 27 · 27046834 + 1 2121310 11 Oct 2018 Andrew M. Farrow (27121 Prime Search) [17]
11 3 · 27033641 + 1 2117338 21 Feb 2011 Michael Herder (321 Prime Search) [18]
12 33661 · 27031232 + 1 2116617 17 Oct 2007 Sturle Sunde (Seventeen or Bust) [10]
13 6679881 · 26679881 + 1 2010852 25 Jul 2009 Yes Magnus Bergman (Cullen Prime Search) [19]
14 1582137 · 26328550 + 1 1905090 20 Apr 2009 Yes Dennis R. Gesker (Cullen Prime Search) [20]
15 7 · 25775996 + 1 1738749 2 Nov 2012 Martyn Elvy (Proth Prime Search) [21]
16 9 · 25642513 + 1 1698567 29 Nov 2013 Serge Batalov [22][23][nb 1]
17 258317 · 25450519 + 1 1640776 28 Jul 2008 Scott Gilvey (Prime Sierpinski Problem) [14][24]
18 27 · 25213635 + 1 1569462 9 Mar 2015 Hiroyuki Okazaki (27121 Prime Search) [25]
19 39 · 25119458 + 1 1541113 23 Nov 2019 Scott Brown (Fermat Divisor Prime Search) [26]
20 3 · 25082306 + 1 1529928 3 Apr 2009 Andy Brady (321 Prime Search) [27]
  1. ^ It remains unclear about which project did Batalov join to find the prime; however, we can be sure that he did not use PrimeGrid.

UsesEdit

Small Proth primes (less than 10200) have been used in constructing prime ladders, sequences of prime numbers such that each term is "close" (within about 1011) to the previous one. Such ladders have been used to empirically verify prime-related conjectures. For example, Goldbach's weak conjecture was verified in 2008 up to 8.875·1030 using prime ladders constructed from Proth primes.[28] (The conjecture was later proved by Harald Helfgott.[29][30][better source needed])

Also, Proth primes can optimize den Boer reduction between the Diffie-Hellman problem and the Discrete logarithm problem. The prime number 55×2286 + 1 has been used in this way.[31]

As Proth primes have simple binary representations, they have also been used in fast modular reduction without the need for pre-computation, for example by Microsoft.[32]


ReferencesEdit

  1. ^ a b c Sze, Tsz-Wo (2008). "Deterministic Primality Proving on Proth Numbers". arXiv:0812.2596 [math.NT].
  2. ^ a b Weisstein, Eric W. "Proth Prime". mathworld.wolfram.com. Retrieved 2019-12-06.
  3. ^ Weisstein, Eric W. "Proth Number". mathworld.wolfram.com. Retrieved 2019-12-07.
  4. ^ Weisstein, Eric W. "Proth's Theorem". MathWorld.
  5. ^ Konyagin, Sergei; Pomerance, Carl (2013), Graham, Ronald L.; Nešetřil, Jaroslav; Butler, Steve (eds.), "On Primes Recognizable in Deterministic Polynomial Time", The Mathematics of Paul Erdős I, Springer New York, pp. 159–186, doi:10.1007/978-1-4614-7258-2_12, ISBN 978-1-4614-7258-2
  6. ^ Caldwell, Chris. "The Top Twenty: Proth". The Prime Pages.
  7. ^ Van Zimmerman (30 Nov 2016) [9 Nov 2016]. "World Record Colbert Number discovered!". PrimeGrid.
  8. ^ Caldwell, Chris. "The Top Twenty: Largest Known Primes". The Prime Pages.
  9. ^ Caldwell, Chris K. "The top twenty: Proth". The Top Twenty. Retrieved 6 December 2019.
  10. ^ a b c d e Goetz, Michael (27 February 2018). "Seventeen or Bust". PrimeGrid. Retrieved 6 Dec 2019.
  11. ^ "Official discovery of the prime number 168451×219375200+1" (PDF). PrimeGrid. Retrieved 6 Dec 2019.
  12. ^ "Official discovery of the prime number 193997×211452891+1" (PDF). PrimeGrid. Retrieved 6 Dec 2019.
  13. ^ "Official discovery of the prime number 3×210829346+1" (PDF). PrimeGrid. Retrieved 6 Dec 2019.
  14. ^ a b "The Prime Sierpinski Problem". mersenneforum.org. 18 Jun 2004. Retrieved 7 December 2019.
  15. ^ Caldwell, Chris K. "Patrice Salah". The Prime Pages. Retrieved 9 December 2019.
  16. ^ "Official discovery of the prime number 161041×27107964+1" (PDF). PrimeGrid. Retrieved 6 Dec 2019.
  17. ^ "Official discovery of the prime number 27×27046834+1" (PDF). PrimeGrid. Retrieved 6 Dec 2019.
  18. ^ "Official discovery of the prime number 3×27033641+1" (PDF). PrimeGrid. Retrieved 6 Dec 2019.
  19. ^ "Official discovery of the prime number 6679881×26679881+1" (PDF). PrimeGrid. Retrieved 6 Dec 2019.
  20. ^ "Official discovery of the prime number 6328548×26328548+1" (PDF). PrimeGrid. Retrieved 6 Dec 2019.
  21. ^ "Official discovery of the prime number 7×25775996+1" (PDF). PrimeGrid. Retrieved 6 Dec 2019.
  22. ^ "Suggestion: a 5-7-9 Proth project". PrimeGrid. 25 Jul 2019. Retrieved 8 Dec 2019.
  23. ^ "9·25642513+1 (Another of the Prime Pages' resources)". The Prime Database. 1 April 2014. Retrieved 9 December 2019.
  24. ^ Caldwell, Chris K. "Scott Gilvey". The Prime Pages. Retrieved 9 December 2019.
  25. ^ "Official discovery of the prime number 27×25213635+1" (PDF). PrimeGrid. Retrieved 6 Dec 2019.
  26. ^ "PrimeGrid Primes". PrimeGrid. Retrieved 7 December 2019.
  27. ^ "Official discovery of the prime number 3×25082306+1" (PDF). PrimeGrid. Retrieved 6 Dec 2019.
  28. ^ Helfgott, H. A.; Platt, David J. (2013). "Numerical Verification of the Ternary Goldbach Conjecture up to 8.875e30". arXiv:1305.3062 [math.NT].
  29. ^ Helfgott, Harald A. (2013). "The ternary Goldbach conjecture is true". arXiv:1312.7748 [math.NT].
  30. ^ "Harald Andrés Helfgott". Alexander von Humboldt-Professur. Retrieved 2019-12-08.
  31. ^ Brown, Daniel R. L. (24 Feb 2015). "CM55: special prime-field elliptic curves almost optimizing den Boer's reduction between Diffie–Hellman and discrete logs" (PDF). International Association for Cryptologic Research: 1–3.
  32. ^ Acar, Tolga; Shumow, Dan (2010). "Modular Reduction without Pre-Computation for Special Moduli" (PDF). Microsoft Research.