A001567 Fermat pseudoprimes to base 2, also called Sarrus numbers or Poulet numbers.

341, 561, 645, 1105, 1387, 1729, 1905, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321, 8481, 8911, 10261, 10585, 11305, 12801, 13741, 13747, 13981, 14491, 15709, 15841, 16705, 18705, 18721, 19951, 23001, 23377, 25761, 29341

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

A composite number n is a Fermat pseudoprime to base b if and only if b^(n-1) == 1 (mod n). Fermat pseudoprimes to base 2 are often simply called pseudoprimes.

Theorem: If both numbers q and 2q - 1 are primes (q is in the sequence A005382) and n = q*(2q-1) then 2^(n-1) == 1 (mod n) (n is in the sequence) if and only if q is of the form 12k + 1. The sequence 2701, 18721, 49141, 104653, 226801, 665281, 721801, ... is related. This subsequence is also a subsequence of the sequences A005937 and A020137. - Farideh Firoozbakht, Sep 15 2006

Also, composite odd numbers n such that n divides 2^n - 2 (cf. A006935). It is known that all primes p divide 2^(p-1) - 1. There are only two known numbers n such that n^2 divides 2^(n-1) - 1, A001220(n) = {1093, 3511} Wieferich primes p: p^2 divides 2^(p-1) - 1. 1093^2 and 3511^2 are the terms of a(n). - Alexander Adamchuk, Nov 06 2006

An odd composite number 2n + 1 is in the sequence if and only if multiplicative order of 2 (mod 2n+1) divides 2n. - Ray Chandler, May 26 2008

The Carmichael numbers A002997 are a subset of this sequence. For the Sarrus numbers which are not Carmichael numbers, see A153508. - Artur Jasinski, Dec 28 2008

An odd number n greater than 1 is a Fermat pseudoprime to base b if and only if ((n-1)! - 1)*b^(n-1) == -1 (mod n). - Arkadiusz Wesolowski, Aug 20 2012

The name "Sarrus numbers" is after Frédéric Sarrus, who, in 1819, discovered that 341 is a counterexample to the "Chinese hypothesis" that n is prime if and only if 2^n is congruent to 2 (mod n). - Alonso del Arte, Apr 28 2013

The name "Poulet numbers" appears in Monografie Matematyczne 42 from 1932, apparently because Poulet in 1928 produced a list of these numbers (cf. Miller, 1975). - Felix Fröhlich, Aug 18 2014

Numbers n > 2 such that (n-1)! + 2^(n-1) == 1 (mod n). Composite numbers n such that (n-2)^(n-1) == 1 (mod n). - Thomas Ordowski, Feb 20 2016

The only twin pseudoprimes up to 10^13 are 4369, 4371. - Thomas Ordowski, Feb 12 2016

Theorem (A. Rotkiewicz, 1965): (2^p-1)*(2^q-1) is a pseudoprime if and only if p*q is a pseudoprime, where p,q are different primes. - Thomas Ordowski, Mar 31 2016

Theorem (W. Sierpiński, 1947): if n is a pseudoprime (odd or even), then 2^n-1 is a pseudoprime. - Thomas Ordowski, Apr 01 2016

If 2^n-1 is a pseudoprime, then n is a prime or a pseudoprime (odd or even). - Thomas Ordowski, Sep 05 2016

From Amiram Eldar, Jun 19 2021, Apr 21 2024: (Start)

Erdős (1950) called these numbers "almost primes".

According to Erdős (1949) and Piza (1954), the term "pseudoprime" was coined by D. H. Lehmer.

Named after the French mathematician Pierre de Fermat (1607-1665), or, alternatively, after the Belgian mathematician Paul Poulet (1887-1946), or, the French mathematician Pierre Frédéric Sarrus (1798-1861). (End)

If m is a term of this sequence, then (m-1)/ord(2,m) >= 5, where ord(a,m) is the multiplicative order of a modulo m; see my link below. Actually, it seems that we always have (m-1)/ord(2,m) >= 9. - Jianing Song, Nov 04 2024

References

• Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, p. 80.
• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A12, pp. 44-50.
• George P. Loweke, The Lore of Prime Numbers. New York: Vantage Press (1982), p. 22.
• Øystein Ore, Number Theory and Its History, McGraw-Hill, 1948.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 88-92.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 145.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..101629 [The pseudoprimes up to 10^12, from Richard Pinch's web site - see links below]
• Jonathan Bayless and Paul Kinlaw, Explicit Bounds for the Sum of Reciprocals of Pseudoprimes and Carmichael Numbers, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.4.
• Jens Bernheiden, Pseudoprimes (Text in German).
• John Brillhart, N. J. A. Sloane, J. D. Swift, Correspondence, 1972.
• Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
• Paul Erdős, On the converse of Fermat's theorem, The American Mathematical Monthly, Vol. 56, No. 9 (1949), pp. 623-624; alternative link.
• Paul Erdős, On almost primes, Amer. Math. Monthly, Vol. 57, No. 6 (1950), pp. 404-407; alternative link.
• Jan Feitsma, The pseudoprimes below 2^64.
• William Galway, Tables of pseudoprimes and related data [Includes a file with pseudoprimes up to 2^64.]
• Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly, Vol. 95, No. 8 (1988), pp. 697-712. [Annotated scanned copy]
• Paul Kinlaw, The reciprocal sums of pseudoprimes and Carmichael numbers, Mathematics of Computation (2023).
• D. H. Lehmer, Guide to Tables in the Theory of Numbers, Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 48.
• D. H. Lehmer, Errata for Poulet's table, Math. Comp., Vol. 25, No. 116 (1971), pp. 944-945.
• D. H. Lehmer, Errata for Poulet's table. [annotated scanned copy]
• Gérard P. Michon, Pseudoprimes.
• J. C. P. Miller, On factorization, with a suggested new approach, Math. Comp., Vol. 29, No. 129 (1975), pp. 155-172. - _Felix Fröhlich_, Aug 18 2014
• Robert Morris, Some observations on the converse of Fermat's theorem, unpublished memorandum, Oct 03 1973.
• Richard Pinch, Pseudoprimes.
• P. A. Piza, Fermat Coefficients, Mathematics Magazine, Vol. 27, No. 3 (1954), pp. 141-146.
• Carl Pomerance & N. J. A. Sloane, Correspondence, 1991.
• Paul Poulet, Tables des nombres composés vérifiant le théorème de Fermat pour le module 2 jusqu'à 100.000.000, Sphinx (Brussels), Vol. 8 (1938), pp. 42-45. [annotated scanned copy]
• Fred Richman, Primality testing with Fermat's little theorem.
• Andrzej Rotkiewicz, Sur les nombres pseudopremiers de la forme MpMq, Elemente der Mathematik, Vol. 20 (1965), pp. 108-109.
• Waclaw Sierpiński, Remarque sur une hypothèse des Chinois concernant les nombres (2^n-2)/n, Colloquium Mathematicum, Vol. 1 (1947), p. 9.
• Waclaw Sierpiński, Elementary Theory of Numbers, Państ. Wydaw. Nauk., Warszawa, 1964, p. 215.
• Jianing Song, Notes on Fermat Pseudoprimes.
• Eric Weisstein's World of Mathematics, Chinese Hypothesis, Fermat Pseudoprime, Poulet Number, and Pseudoprime.
• Wikipedia, Chinese hypothesis and Pseudoprime.
• Index entries for sequences related to pseudoprimes

Crossrefs

Cf. A002997, A005382, A020137, A052155, A083737, A084653, A153508.

Cf. A001220 = Wieferich primes p: p^2 divides 2^(p-1) - 1.

Cf. A005935, A005936, A005937, A005938, A005939, A020136-A020228 (pseudoprimes to bases 3 through 100).

Programs

Magma

[n: n in [3..3*10^4 by 2] | IsOne(Modexp(2,n-1,n)) and not IsPrime(n)]; // Bruno Berselli, Jan 17 2013

Maple

select(t -> not isprime(t) and 2 &^(t-1) mod t = 1, [seq(i,i=3..10^5,2)]); # Robert Israel, Feb 18 2016

Mathematica

Select[Range[3,30000,2], ! PrimeQ[ # ] && PowerMod[2, (# - 1), # ] == 1 &] (* Farideh Firoozbakht, Sep 15 2006 *)

PARI

q=1;vector(50,i,until( !isprime(q) & (1<<(q-1)-1)%q == 0, q+=2);q) \\ M. F. Hasler, May 04 2007

PARI

is_A001567(n)={Mod(2,n)^(n-1)==1 && !isprime(n) && n>1}  \\ M. F. Hasler, Oct 07 2012, updated to current PARI syntax and to exclude even pseudoprimes on Mar 01 2019

Formula

Sum_{n>=1} 1/a(n) is in the interval (0.015260, 33) (Bayless and Kinlaw, 2017). The upper bound was reduced to 0.0911 by Kinlaw (2023). - Amiram Eldar, Oct 15 2020, Feb 24 2024

Extensions

More terms from David W. Wilson, Aug 15 1996

Replacement of broken geocities link by Jason G. Wurtzel, Sep 05 2010

"Poulet numbers" added to name by Joerg Arndt, Aug 18 2014