Pillai Prime
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In
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, a Pillai prime is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
''p'' for which there is an integer ''n'' > 0 such that the
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...
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,
139 139 may refer to: * 139 (number), an integer * AD 139, a year of the Julian calendar * 139 BC, a year of the pre-Julian Roman calendar * 139 (New Jersey bus) * 139 Juewa 139 Juewa ( ) is a very large and dark main belt asteroid. It is probabl ...
, 149,
193 Year 193 ( CXCIII) was a common year starting on Monday of the Julian calendar. At the time, it was known as the Year of the Consulship of Sosius and Ericius (or, less frequently, year 946 ''Ab urbe condita''). The denomination 193 for this ye ...
, ... Pillai primes are named after the mathematician
Subbayya Sivasankaranarayana Pillai Subbayya Sivasankaranarayana Pillai (5 April 1901 – 31 August 1950) was an Indian mathematician specialising in number theory. His contribution to Waring's problem was described in 1950 by K. S. Chandrasekharan as "almost certainly his best ...
, who studied these numbers. Their infinitude has been proven several times, by
Subbarao Subbarao or Subba Rao (, Kannada: ಸುಬ್ಬ ರಾವ್), sometimes spelled SubbaRow or Subba Row, is an Indian name. People with the name

* Adurthi Subba Rao (1912–1975), Telugu film director * B. A. Subba Rao (fl. 1950–1987), Telug ...
, Erdős, and Hardy & Subbarao.


References

*. *. *https://planetmath.org/pillaiprime,
PlanetMath PlanetMath is a free content, free, collaborative, mathematics online encyclopedia. Intended to be comprehensive, the project is currently hosted by the University of Waterloo. The site is owned by a US-based nonprofit corporation, "PlanetMath.org ...
Classes of prime numbers Eponymous numbers in mathematics Factorial and binomial topics {{numtheory-stub