A Pillai prime is a prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

 p for which there is an integer n > 0 such that the factorial

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

 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, but . The first few Pillai primes are

23

23 (number)

23 is the natural number following 22 and preceding 24.- In mathematics :Twenty-three is the ninth prime number, the smallest odd prime that is not a twin prime. Twenty-three is also the fifth factorial prime, the third Woodall prime...

, 29

29 (number)

29 is the natural number following 28 and preceding 30.-In mathematics:It is the tenth prime number, and also the fourth primorial prime. It forms a twin prime pair with thirty-one, which is also a primorial prime. Twenty-nine is also the sixth Sophie Germain prime. It is also the sum of three...

, 59

59 (number)

59 is the natural number following 58 and preceding 60.-In mathematics:Fifty-nine is the 17th smallest prime number. The next is sixty-one, with which it comprises a twin prime. 59 is an irregular prime, a safe prime and the 14th supersingular prime. It is an Eisenstein prime with no imaginary...

, 61

61 (number)

61 is the natural number following 60 and preceding 62.-In mathematics:It is the 18th prime number. The previous is 59, with which it comprises a twin prime. Sixty-one is a cuban prime of the form p = / , x = y + 1....

, 67

67 (number)

67 is the natural number following 66 and preceding 68. It is an odd number.-In mathematics:Sixty-seven is the 19th prime number , an irregular prime, a lucky prime, the sum of five consecutive primes , and a Heegner number.Since 18! + 1 is divisible by 67 but 67 is not one more than a multiple of...

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 79

79 (number)

Seventy-nine is the natural number following 78 and preceding 80.79 may represent:-In mathematics:*An odd number*The smallest number that can't be represented as a sum of fewer than 19 fourth powers*A strictly non-palindromic number...

, 83

83 (number)

83 is the natural number following 82 and preceding 84.-In mathematics:Eighty-three is the sum of three consecutive primes as well as the sum of five consecutive primes ....

, 109

109 (number)

109 is the natural number following 108 and preceding 110.-In mathematics:One hundred [and] nine is the 29th prime number, and also a Chen prime. The previous prime is 107, making them both twin primes...

, 137

137 (number)

137 is the natural number following 136 and preceding 138.-In mathematics :One hundred [and] thirty-seven is the 33rd prime number; the next is 139, with which it comprises a twin prime, and thus 137 is a Chen prime. 137 is an Eisenstein prime with no imaginary part and a real part of the form 3n -...

, 139

139 (number)

139 is the natural number following 138 and preceding 140.-In mathematics:One hundred [and] thirty-nine is the 34th prime number, so it is divisible only by itself and 1. It is a twin prime with 137. Because 141 is a semiprime, 139 is a Chen prime...

, 149

149 (number)

149 is the natural number between 148 and 150. It is also a prime number.-In mathematics:*149 is the 35th prime number, and with the next prime number, 151, is a twin prime, thus 149 is a Chen prime. 149 is a strong prime in the sense that it is more than the arithmetic mean of its two neighboring...

, 193

193 (number)

193 is the natural number following 192 and preceding 194.-In mathematics:* 193 is an odd number* 193 is a centered 32-gonal number* 193 is a deficient number, as 1 is less than 193* 193 is a happy number* 193 is a lucky number...

, ...

Pillai primes are named after the mathematician Subbayya Sivasankaranarayana Pillai

Subbayya Sivasankaranarayana Pillai

Subbayya Sivasankaranarayana Pillai was an Indian mathematician, well known for his work in number theory. He was from Tamil Nadu....

, who asked about these numbers. Their infinitude has been proved several times, by Subbarao, Erdős, and Hardy & Subbarao.