# Proth prime

Named after François Proth 1878 Proth, Francois Over 1.5 billion below 270 Infinite Proth numbers, prime numbers k × 2n + 1 3, 5, 13, 17, 41, 97, 113 10223 × 231172165 + 1 (as of December 2019) A080076Proth 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 $N=k\cdot 2^{n}+1$ where k and n are positive integers, k is odd and $2^{n}>k$ . A Proth prime is a Proth number that is prime. They are named after the French mathematician François Proth. 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 ().

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

## Definition

A Proth number takes the form $N=k2^{n}+1$ where k and n are positive integers, $k$ is odd and $2^{n}>k$ . A Proth prime is a Proth number that is prime.

Without the condition that $2^{n}>k$ , all odd integers larger than 1 would be Proth numbers.

## Primality testing

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

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

## Large primes

As of 2019, the largest known Proth prime is $10223\cdot 2^{31172165}+1$ . It is 9,383,761 digits long. It was found by Szabolcs Peter in the PrimeGrid distributed computing project which announced it on 6 November 2016. It is also the largest known non-Mersenne prime.

The project Seventeen or Bust, searching for Proth primes with a certain $t$ 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 $3\times 2^{n}+1$ , also called Thabit primes of the second kind)
• 27121 Prime Search (searching for primes of the form $27\times 2^{n}+1$ and $121\times 2^{n}+1$ )
• Cullen prime search (searching for primes of the form $n\times 2^{n}+1$ )
• Sierpinski problem (and their prime and extended generalizations) – searching for primes of the form $k\times 2^{n}+1$ 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:

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

## 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. (The conjecture was later proved by Harald Helfgott.[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.

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.