Proth prime

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Short description: Prime number of the form k*(2^n)+1
Proth prime
Named afterFrançois Proth
Publication year1878
Author of publicationProth, Francois
No. of known terms4304683178 below 272 [1]
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

A Proth number is a natural number N of the form [math]\displaystyle{ N = k \times 2^n +1 }[/math] where k and n are positive integers, k is odd and [math]\displaystyle{ 2^n \gt k }[/math]. A Proth prime is a Proth number that is prime. They are named after the French mathematician François Proth.[2] The first few Proth primes are

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 (OEIS: A080076).

It is still an open question whether an infinite number of Proth primes exist. It was shown in 2022 that the reciprocal sum of Proth primes converges to a real number near 0.747392479, substantially less than the value of 1.093322456 for the reciprocal sum of Proth numbers.[1]

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

Definition

A Proth number takes the form [math]\displaystyle{ N=k 2^n +1 }[/math] where k and n are positive integers, [math]\displaystyle{ k }[/math] is odd and [math]\displaystyle{ 2^n\gt k }[/math]. A Proth prime is a Proth number that is prime.[2][3] Without the condition that [math]\displaystyle{ 2^n \gt k }[/math], all odd integers larger than 1 would be Proth numbers.[4]

Primality testing

The primality of a Proth number can be tested with Proth's theorem, which states that a Proth number [math]\displaystyle{ p }[/math] is prime if and only if there exists an integer [math]\displaystyle{ a }[/math] for which

[math]\displaystyle{ a^{\frac{p-1}{2}}\equiv -1 \pmod{p}. }[/math][3][5]

This theorem can be used as a probabilistic test of primality, by checking for many random choices of [math]\displaystyle{ a }[/math] whether [math]\displaystyle{ a^{\frac{p-1}{2}}\equiv -1 \pmod{p}. }[/math] If this fails to hold for several random [math]\displaystyle{ a }[/math], then it is very likely that the number [math]\displaystyle{ p }[/math] 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 [math]\displaystyle{ \tilde{O}((k\log k+\log N)(\log N)^2) }[/math] time, where Õ is the soft-O notation. For typical searches for Proth primes, usually [math]\displaystyle{ k }[/math] is either fixed (e.g. 321 Prime Search or Sierpinski Problem) or of order [math]\displaystyle{ O(\log N) }[/math] (e.g. Cullen prime search). In these cases algorithm runs in at most [math]\displaystyle{ \tilde{O}((\log N)^3) }[/math], or [math]\displaystyle{ O((\log N)^{3+\epsilon}) }[/math] time for all [math]\displaystyle{ \epsilon\gt 0 }[/math]. There is also an algorithm that runs in [math]\displaystyle{ \tilde{O}((\log N)^{24/7}) }[/math] time.[2][6]

Large primes

(As of 2022), the largest known Proth prime is [math]\displaystyle{ 10223 \times 2^{31172165} + 1 }[/math]. It is 9,383,761 digits long.[7] It was found by Szabolcs Peter in the PrimeGrid volunteer computing project which announced it on 6 November 2016.[8] It is also the second largest known non-Mersenne prime.[9]

The project Seventeen or Bust, searching for Proth primes with a certain [math]\displaystyle{ t }[/math] to prove that 78557 is the smallest Sierpinski number (Sierpinski problem), has found 11 large Proth primes by 2007. Similar resolutions to the prime Sierpiński problem and extended Sierpiński problem have yielded several more numbers.

Since divisors of Fermat numbers [math]\displaystyle{ F_n = 2^{2^n} + 1 }[/math] are always of the form [math]\displaystyle{ k \times 2^{n+2} + 1 }[/math], it is customary to determine if a new Proth prime divides a Fermat number.[10]

As of July 2023, 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 [math]\displaystyle{ 3\times2^n+1 }[/math], also called Thabit primes of the second kind)
  • 27121 Prime Search (searching for primes of the form [math]\displaystyle{ 27\times2^n+1 }[/math] and [math]\displaystyle{ 121\times2^n+1 }[/math])
  • Cullen prime search (searching for primes of the form [math]\displaystyle{ n\times2^n+1 }[/math])
  • Sierpinski problem (and their prime and extended generalizations) – searching for primes of the form [math]\displaystyle{ k \times 2^n+1 }[/math] where k is in this list:

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

As of June 2023, the largest Proth primes discovered are:[11]

rank prime digits when Comments Discoverer (Project) References
1 10223 × 231172165 + 1 9383761 31 Oct 2016 Szabolcs Péter (Sierpinski Problem) [12]
2 202705 × 221320516 + 1 6418121 1 Dec 2021 Pavel Atnashev (Extended Sierpinski Problem) [13]
3 81 × 220498148 + 1 6170560 13 Jul 2023 Generalized Fermat F2(3 × 25124537) Ryan Propper (LLR) [11]
4 7 × 220267500 + 1 6101127 21 Jul 2022 Divides F20267499(12) Ryan Propper (LLR) [11][14]
5 168451 × 219375200 + 1 5832522 17 Sep 2017 Ben Maloney (Prime Sierpinski Problem) [15]
6 7 × 218233956 + 1 5488969 1 Oct 2020 Divides Fermat F18233954 and F18233952(7) Ryan Propper [16][14]
7 13 × 216828072 + 1 5065756 11 Oct 2023 Ryan Propper [11]
8 3 × 216408818 + 1 4939547 28 Oct 2020 Divides F16408814(3), F16408817(5), and F16408815(8) James Brown (PrimeGrid) [14]
9 11 × 215502315 + 1 4666663 8 Jan 2023 Divides F15502313(10) Ryan Propper [14]
10 37 × 215474010 + 1 4658143 8 Nov 2022 Ryan Propper [14]
11 (27658613 + 1) × 27658614 + 1 4610945 31 Jul 2020 Gaussian Mersenne norm Ryan Propper and Serge Batalov [11]
12 13 × 215294536 + 1 4604116 30 Sep 2023 Ryan Propper [11]
13 37 × 214166940 + 1 4264676 24 Jun 2022 Ryan Propper [11]
14 99739 × 214019102 + 1 4220176 24 Dec 2019 Brian Niegocki (Extended Sierpinski Problem) [17]
15 404849 × 213764867 + 1 4143644 10 Mar 2021 Generalized Cullen with base 131072 Ryan Propper and Serge Batalov [11]
16 25 × 213719266 + 1 4129912 21 Sep 2022 F1(5 × 26859633) Ryan Propper [11]
17 81 × 213708272 + 1 4126603 11 Oct 2022 F2(3 × 23427068) Ryan Propper [11]
18 81 × 213470584 + 1 4055052 9 Oct 2022 F2(3 × 23367646) Ryan Propper [11]
19 9 × 213334487 + 1 4014082 31 Mar 2020 Divides F13334485(3), F13334486(7), and F13334484(8) Ryan Propper [14]
20 19249 × 213018586 + 1 3918990 26 Mar 2007 Konstantin Agafonov (Seventeen or Bust) [12]

Uses

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.[18] (The conjecture was later proved by Harald Helfgott.[19][20][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.[21]

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.[22]

References

  1. ↑ 1.0 1.1 Borsos, Bertalan; Kovács, Attila; Tihanyi, Norbert (2022), "Tight upper and lower bounds for the reciprocal sum of Proth primes" (in en), Ramanujan Journal (Springer) 59: 181–198, doi:10.1007/s11139-021-00536-2 
  2. ↑ 2.0 2.1 2.2 Sze, Tsz-Wo (2008). "Deterministic Primality Proving on Proth Numbers". arXiv:0812.2596 [math.NT].
  3. ↑ 3.0 3.1 Weisstein, Eric W.. "Proth Prime" (in en). http://mathworld.wolfram.com/ProthPrime.html. 
  4. ↑ Weisstein, Eric W.. "Proth Number" (in en). http://mathworld.wolfram.com/ProthNumber.html. 
  5. ↑ Weisstein, Eric W.. "Proth's Theorem". http://mathworld.wolfram.com/ProthsTheorem.html. 
  6. ↑ Konyagin, Sergei; Pomerance, Carl (2013), Graham, Ronald L.; Nešetřil, Jaroslav; Butler, Steve, eds., "On Primes Recognizable in Deterministic Polynomial Time" (in en), 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 
  7. ↑ Caldwell, Chris. "The Top Twenty: Proth". The Prime Pages. http://primes.utm.edu/top20/page.php?id=66. 
  8. ↑ Van Zimmerman (30 Nov 2016). "World Record Colbert Number discovered!". PrimeGrid. http://www.primegrid.com/forum_thread.php?id=7116. 
  9. ↑ Caldwell, Chris. "The Top Twenty: Largest Known Primes". The Prime Pages. http://primes.utm.edu/top20/page.php?id=3. 
  10. ↑ "The Prime Glossary: Fermat divisor". https://primes.utm.edu/glossary/xpage/FermatDivisor.html. 
  11. ↑ 11.00 11.01 11.02 11.03 11.04 11.05 11.06 11.07 11.08 11.09 11.10 Caldwell, Chris K.. "The top twenty: Proth". https://primes.utm.edu/top20/page.php?id=66. 
  12. ↑ 12.0 12.1 Goetz, Michael (27 February 2018). "Seventeen or Bust". http://primegrid.com/forum_thread.php?id=1647. 
  13. ↑ "PrimeGrid's Extended Sierpinski Problem Prime Search". PrimeGrid. http://www.primegrid.com/download/ESP-202705.pdf. 
  14. ↑ 14.0 14.1 14.2 14.3 14.4 14.5 "New GFN factors". http://www.prothsearch.com/GFNfacs.html. 
  15. ↑ "Official discovery of the prime number 168451×219375200+1". http://www.primegrid.com/download/PSP_168451.pdf. 
  16. ↑ "Fermat factoring status". http://www.prothsearch.com/fermat.html. 
  17. ↑ "Official discovery of the prime number 99739×214019102+1". 24 December 2019. https://www.primegrid.com/download/ESP-99739.pdf. 
  18. ↑ Helfgott, H. A.; Platt, David J. (2013). "Numerical Verification of the Ternary Goldbach Conjecture up to 8.875e30". arXiv:1305.3062 [math.NT].
  19. ↑ Helfgott, Harald A. (2013). "The ternary Goldbach conjecture is true". arXiv:1312.7748 [math.NT].
  20. ↑ "Harald Andrés Helfgott". https://www.humboldt-professur.de/en/preistraeger/preistraeger-2015/helfgott-harald-andres. 
  21. ↑ 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". International Association for Cryptologic Research: 1–3. https://eprint.iacr.org/2014/877.pdf. 
  22. ↑ Acar, Tolga; Shumow, Dan (2010). "Modular Reduction without Pre-Computation for Special Moduli". Microsoft Research. https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/modmul_no_precomp.pdf. 



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