Lucas–Lehmer–Riesel test

The algorithm is very similar to the Lucas–Lehmer test, but with a variable starting point depending on the value of {{mvar|k}}.

Define a sequence {{math|1=u_i}} for all {{math|1=i > 0}} by:

: $u\_i\; =\; u\_\{i-1\}^2-2.$

Then {{math|1=N = k · 2ⁿ − 1}}, with {{math|1=k < 2ⁿ}}, is prime if and only if it divides {{math|1=u_n−2}}.

The starting value {{math|1=u₀}} is determined as follows.

• If {{math|1=k ≡ 1 or 5 (mod 6)}}: if {{math|1=1 (mod 6)}} and {{mvar|n}} is even, or {{math|1=5 (mod 6)}} and {{mvar|n}} is odd, then {{math|1=3}} divides {{mvar|N}}, and there is no need to test. Otherwise, {{math|1=N ≡ 7 (mod 24)}} and the Lucas sequence {{math|1=V(4,1)}} may be used: we take $u\_0\; =\; (2+\backslash sqrt\{3\})^k+(2-\backslash sqrt\{3\})^k$, which is the {{mvar|k}}th term of that sequence. This is a generalization of the ordinary Lucas–Lehmer test, and reduces to it when {{math|1=k = 1}}.
• Otherwise, we are in the case where {{mvar|k}} is a multiple of {{math|1=3}}, and it is more difficult to select the right value of {{math|1=u₀}}. It is known that if {{math|1=k = 3}} and {{math|1=n ≡ 0 or 3 (mod 4)}}, then we can take {{math|1=u₀ = 5778}}.

An alternative method for finding the starting value {{math|1=u₀}} is given in Rödseth 1994.{{cite journal|last=Rödseth|first=Öystein J.|title=A note on primality tests for {{math|1=N = h · 2ⁿ − 1}}|journal=BIT Numerical Mathematics|volume=34|number=3|date=1994|pages=451–454|doi=10.1007/BF01935653|s2cid=120438959 |url=http://folk.uib.no/nmaoy/papers/luc.pdf|archive-url=https://web.archive.org/web/20160306082833/http://folk.uib.no/nmaoy/papers/luc.pdf|archive-date=March 6, 2016|url-status=dead}} The selection method is much easier than that used by Riesel for the 3-divides-{{mvar|k}} case: first, find a {{mvar|P}}-value that satisfies the following equalities of Jacobi symbols:

:$\backslash left(\backslash frac\{P-2\}\{N\}\backslash right)=1\; \backslash quad\backslash text\{and\}\backslash quad\; \backslash left(\backslash frac\{P+2\}\{N\}\backslash right)=-1.$

In practice, only a few {{mvar|P}}-values need be checked before one is found (5, 8, 9, or 11 work in about 85% of trials).{{citation needed|date=September 2017}}

To find the starting value {{math|1=u₀}} from the {{mvar|P}} value, we can use a Lucas {{math|1=(P,1)}} sequence, as shown in as well as page 124 of.{{cite book | title=Prime Numbers and Computer Methods for Factorization | edition=2nd | first=Hans | last=Riesel | author-link=Hans Riesel | series=Progress in Mathematics | volume=126 | publisher=Birkhäuser | year=1994 | isbn=0-8176-3743-5 | pages=107–121 }} The latter explains that when {{math|1=3 ∤ k}}, {{math|1=P = 4}} may be used as above, and no further search is necessary.

The starting value {{math|1=u₀}} will be the Lucas sequence term {{math|1=V_k(P,1)}} taken modulo {{mvar|N}}. This process of selection takes very little time compared to the main test.

The Lucas–Lehmer–Riesel test is a particular case of group-order primality testing; we demonstrate that some number is prime by showing that some group has the order that it would have were that number prime, and we do this by finding an element of that group of precisely the right order.

For Lucas-style tests on a number {{mvar|N}}, we work in the multiplicative group of a quadratic extension of the integers modulo {{mvar|N}}; if {{mvar|N}} is prime, then the order of this multiplicative group is {{math|1=N² − 1}}, it has a subgroup of order {{math|1=N + 1}}, and we try to find a generator for that subgroup.

We start off by trying to find a non-iterative expression for the {{math|1=u_i}}. Following the model of the Lucas–Lehmer test, put {{math|1=u_i = a²ⁱ + a⁻²ⁱ}}, and by induction we have {{math|1=u_i = u{{su|p=2|b=i−1}} − 2}}.

So we can consider ourselves as looking at the {{math|1=2ⁱ}}th term of the sequence {{math|1=v(i) = aⁱ + aⁱ}}. If {{mvar|a}} satisfies a quadratic equation, then this is a Lucas sequence, and has an expression of the form {{math|1=v(i) = α v(i−1) + β v(i−2)}}. Really, we are looking at the {{math|1=k · 2ⁱ}}th term of a different sequence, but since decimations (take every {{mvar|k}}th term starting with the zeroth) of a Lucas sequence are themselves Lucas sequences, we can deal with the factor {{mvar|k}} by picking a different starting point.

LLR is a program that can run the LLR tests. The program was developed by Jean Penné. Vincent Penné has modified the program so that it can obtain tests via the Internet.{{cite web|url=https://mersenneforum.org/showthread.php?t=13165|title=LLRnet supports LLR V3.8! (LLRnet2010 V0.73L)|last=Bonath|first=Karsten|date=2010-03-12|work=Great Internet Mersenne Prime Search forum|access-date=17 November 2021}} The software is used by both individual prime searchers and some distributed computing projects including Riesel Sieve and PrimeGrid.

A revised version, LLR2{{cite web |url=https://github.com/patnashev/llr2 |title=LLR2 GitHub |first=Pavel |last=Atnashev |website=GitHub |access-date=2023-11-23}} was deployed in 2020.{{cite web |url=https://www.primegrid.com/forum_thread.php?id=9303 |title=LLR2 installed on all big LLR projects |website=PrimeGrid message boards |date=11 September 2020 |access-date=2023-11-23}} This generates a "proof of work" certificate which allows the computation to be verified without needing a full double-check.

A further update, PRST{{cite web |url=https://github.com/patnashev/prst |title=PRST GitHub |first=Pavel |last=Atnashev |website=GitHub |access-date=2023-11-23}} uses an alternate certificate scheme{{cite arXiv |eprint=2209.15623 |class=cs.CR |title=An Efficient Modular Exponentiation Proof Scheme |first1=Darren |last1=Li |first2=Yves |last2=Gallot |date=16 September 2022}} which takes longer to verify but is faster to generate for some forms of prime.{{cite web |url=https://www.primegrid.com/forum_thread.php?id=10296 |title=SR5 project switched to PRST |website=PrimeGrid message boards |date=19 July 2023 |access-date=2023-11-23}}

• Riesel number

{{reflist}}

• {{cite journal |last=Riesel |first=Hans |year=1969 |title=Lucasian Criteria for the Primality of N = h·2ⁿ − 1 |journal=Mathematics of Computation |volume=23 |issue=108 |pages=869–875 |doi=10.2307/2004975 |jstor= 2004975|publisher=American Mathematical Society }}

• [http://jpenne.free.fr/index2.html Download Jean Penné's LLR]
• [https://github.com/danaj/Math-Prime-Util-GMP Math::Prime::Util::GMP] - Part of Perl's ntheory module, this has basic implementations of LLR and Proth form testing, as well as some BLS75 proof methods.

{{Number-theoretic algorithms}}

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Category:Primality tests