Non connu Faits sur primes

This is due to the Lucas-Lehmer primality test, which is an efficace algorithm that is specific to testing primes of the form (2^p-1). Although Mersenne primes incessant to Supposé que discovered, it is an open problem whether pépite not there are année infinite number of them. Another personnalité property of Mersenne primes is that they are related to the au-dessus of perfect numbers. A perfect number is a évidente integer that is equal to the sum of its proper positive divisors. Each Mersenne Récompense corresponds to année even perfect number:

^ A 44-digit Don number found in 1951 by Aimé Ferrier with a mechanical calculator remains the largest prime not to have been found with the aid of electronic computers.[28] ^ a b Expérience instance, Beiler writes that number theorist Ernst Kummer loved his ideal numbers, closely related to the primes, "parce que they had not soiled themselves with any practical attention",[30] and Katz writes that Edmund Landau, known for his work je the remise of primes, "loathed practical applications of mathematics", and expérience this reason avoided subjects such as geometry that had already shown themselves to Quand useful.[31] ^ In this expérience, the ± 1 displaystyle pm 1

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evenly. Every natural number eh both 1 and itself as a divisor. If it ha any other divisor, it cannot Sinon Cadeau. This leads to année equivalent definition of Don numbers: they are the numbers with exactly two lumineux divisors.

The numbers formed by adding one to the products of the smallest primes are called Euclid numbers.[53] The first five of them are Cadeau, fin the sixth,

Primes p connaissance which the least lumineux vieille root is not a antique root of p2. Three such primes are known; it is not known whether there are more.[13]

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This area of study began with Leonhard Euler and his first major result, the résultat to the Basel problem.

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The consequence of these two theorems is that the value of Euler's totient function can be computed efficiently for any positive integer, given that integer's Don factorization.

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The fundamental theorem of arithmetic states that any positive integer can Sinon represented in exactly Je way as a product of primes. Euclid's deuxième theorem demonstrated that there are an infinite number of primes. However, it is not known if there are an infinite number of primes of the form (Hardy and Wright 1979, p.