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In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: 7 = 7 1 , 9 = 3 2 and 64 = 2 6 are prime powers, while 6 = 2 × 3 , 12 = 2 2 × 3 and 36 = 6 2 = 2 2 × 3 2 are not.
A plot of the number of digits in the largest known prime by year, since the electronic computer. The vertical scale is logarithmic. A prime number is a natural number greater than 1 with no divisors other than 1 and itself. According to Euclid's theorem there are infinitely many prime numbers, so there is no largest prime.
However, it does not contain all the prime numbers, since the terms gcd(n + 1, a n) are always odd and so never equal to 2. 587 is the smallest prime (other than 2) not appearing in the first 10,000 outcomes that are different from 1. Nevertheless, in the same paper it was conjectured to contain all odd primes, even though it is rather inefficient.
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form M n = 2 n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If n is a composite number then so is 2 n − 1.
As one special case, it can be used to prove that if n is a positive integer then 4 divides () if and only if n is not a power of 2. It follows from Legendre's formula that the p -adic exponential function has radius of convergence p − 1 / ( p − 1 ) {\displaystyle p^{-1/(p-1)}} .
If n is a power of an odd prime number the formula for the totient says its totient can be a power of two only if n is a first power and n − 1 is a power of 2. The primes that are one more than a power of 2 are called Fermat primes, and only five are known: 3, 5, 17, 257, and 65537. Fermat and Gauss knew of these.
Riemann's prime-power counting function is usually denoted as Π 0 (x) or J 0 (x). It has jumps of 1 / n at prime powers p n and it takes a value halfway between the two sides at the discontinuities of π(x). That added detail is used because the function may then be defined by an inverse Mellin transform. Formally, we may define Π 0 ...
In mathematics, Kummer's theorem is a formula for the exponent of the highest power of a prime number p that divides a given binomial coefficient. In other words, it gives the p-adic valuation of a binomial coefficient. The theorem is named after Ernst Kummer, who proved it in a paper, (Kummer 1852).