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An illegal prime is an illegal number which is also prime.One of the earliest illegal prime numbers was generated in March 2001 by Phil Carmody.Its binary representation corresponds to a compressed version of the C source code of a computer program implementing the DeCSS decryption algorithm, which can be used by a computer to circumvent a DVD's copy protection.
This is a list of articles about prime numbers. A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes.
The reciprocals of prime numbers have been of interest to mathematicians for various reasons. They do not have a finite sum , as Leonhard Euler proved in 1737. Like rational numbers , the reciprocals of primes have repeating decimal representations.
The table below lists the largest currently known prime numbers and probable primes (PRPs) as tracked by the PrimePages and by Henri & Renaud Lifchitz's PRP Records. Numbers with more than 2,000,000 digits are shown.
The following table lists the progression of the largest known prime number in ascending order. [3] Here M p = 2 p − 1 is the Mersenne number with exponent p, where p is a prime number. The longest record-holder known was M 19 = 524,287, which was the largest known prime for 144 years. No records are known prior to 1456.
The smallest n-digit number to achieve this number of primes is 2, 37, 137, 1379, 13679, 123479, 1234679, 12345679, 102345679, 1123456789, 10123456789, ... (sequence A134596 in the OEIS) Primeval numbers can be composite. The first is 1037 = 17×61. A Primeval prime is a primeval number which is also a prime number:
Not all Euclid numbers are prime. E 6 = 13# + 1 = 30031 = 59 × 509 is the first composite Euclid number. Every Euclid number is congruent to 3 modulo 4 since the primorial of which it is composed is twice the product of only odd primes and thus congruent to 2 modulo 4. This property implies that no Euclid number can be a square.
In mathematics, a primorial prime is a prime number of the form p n # ± 1, where p n # is the primorial of p n (i.e. the product of the first n primes). [1] Primality tests show that: p n # − 1 is prime for n = 2, 3, 5, 6, 13