Pillai prime

Top

Home > Library > Miscellaneous > Wikipedia

A Pillai prime is a prime number p for which there is an integer n > 0 such that the factorial of n is one less than a multiple of the prime, but the prime is not one more than a multiple of n. To put it algebraically, $n! \equiv -1 \mod p$ but $p \not\equiv 1 \mod n$ . The first few Pillai primes are

23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, ... (sequence A063980 in OEIS)

Pillai primes are named after the mathematician Subbayya Sivasankaranarayana Pillai, who asked about these numbers. Their infinitude has been proved several times, by Subbarao, Erdős, and Hardy & Subbarao.

References

Guy, R. K. (2004), Unsolved Problems in Number Theory (3rd ed.), New York: Springer-Verlag, p. A2, ISBN 0-387-20860-7 .
Hardy, G. E. & Subbarao, M. V. (2002), "A modified problem of Pillai and some related questions", American Mathematical Monthly 109 (6): 554–559, doi:10.2307/2695445 .
Pillai prime, PlanetMath.org.

v t e Prime number classes

By formula	Fermat (2^2ⁿ+1) · Mersenne (2^p−1) · Double Mersenne (2^{2^p−1}−1) · Wagstaff (2^p+1)/3 · Proth (k·2ⁿ+1) · Factorial (n!±1) · Primorial (p_n#±1) · Euclid (p_n#+1) · Pythagorean (4n+1) · Pierpont (2^u·3^v+1) · Solinas (2^a±2^b±1) · Cullen (n·2ⁿ+1) · Woodall (n·2ⁿ−1) · Cuban (x³−y³)/(x−y) · Carol (2ⁿ−1)²−2 · Kynea (2ⁿ+1)²−2 · Leyland (x^y+y^x) · Thabit (3·2ⁿ−1) · Mills (floor(A^3ⁿ))

By integer sequence	Fibonacci · Lucas · Motzkin · Bell · Partitions · Pell · Perrin · Newman-Shanks-Williams

By property	Lucky · Wall-Sun-Sun · Wilson · Wieferich · Wieferich pair · Fortunate · Ramanujan · Pillai · Regular · Strong · Stern · Supersingular (elliptic curve) · Supersingular (moonshine theory) · Wolstenholme · Good · Super · Higgs · Highly cototient · Illegal

Base-dependent	Happy · Dihedral · Palindromic · Emirp · Repunit · Permutable · Circular · Strobogrammatic · Minimal · Full reptend · Unique · Primeval · Self · Smarandache–Wellin

Patterns	Twin (p, p+2) · Triplet (p, p+2 or p+4, p+6) · Quadruplet (p, p+2, p+6, p+8) · Tuple · Cousin (p, p+4) · Sexy (p, p+6) · Chen · Sophie Germain (p, 2p+1) · Cunningham chain (p, 2p±1, …) · Safe (p, (p−1)/2) · Arithmetic progression (p+a·n, n=0,1, …) · Balanced (consecutive p−n, p, p+n)

By size	Titanic (1,000+ digits) · Gigantic (10,000+) · Mega (1,000,000+) · Largest known

Complex numbers	Eisenstein prime · Gaussian prime

Composite numbers	Pseudoprime · Almost prime · Semiprime · Interprime

Related topics	Probable prime · Industrial-grade prime · Formula for primes · Prime gap

List of prime numbers

This number theory-related article is a stub. You can help Wikipedia by expanding it.

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

Help us answer these

How do you get prime?

Is prime factors the same as prime numbers?

What is prime implicant and essential prime implicant?

Example program of prime and not prime numbers?

Post a question - any question - to the WikiAnswers community:

Copyrights:

Cite

Wikipedia on Answers.com This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article Pillai prime. Read