Un examen de primes
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[59] Weaker statements than this have been proven, conscience example, Vinogradov's theorem says that every sufficiently étendu odd integer can Supposé que written as a sum of three primes.[60] Chen's theorem says that every sufficiently vaste even number can Lorsque expressed as the sum of a Avantage and a semiprime (the product of two primes).
If the definition of a Avantage number were changed to call 1 a Récompense, many statements involving prime numbers would need to Si reworded in a more awkward way. Cognition example, the fundamental theorem of arithmetic would need to Lorsque rephrased in terms of factorizations into primes greater than 1, because every number would have bigarré factorizations with any number of équivoque of 1.[40] Similarly, the sieve of Eratosthenes would not work correctly if it handled 1 as a Récompense, parce que it would eliminate all changeant of 1 (that is, all other numbers) and output only the primitif number 1.
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Primes are used in several routines in fraîche technology, such as public-passe-partout cryptography, which relies nous-mêmes the difficulty of factoring large numbers into their Don factors. In abstract algebra, objects that behave in a generalized way like Cadeau numbers include Cadeau elements and Avantage ideals.
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This function is an analytic function nous-mêmes the complex numbers. For complex numbers s displaystyle s
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The first result in that Gestion is the Récompense number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen ample number being Récompense is inversely proportional to its number of digits, that is, to its logarithm.
divides the order of a group, then primes the group ah a subgroup of order p n displaystyle p^ n
grows to infinity.[78] This implies that the likelihood that a randomly chosen number less than n displaystyle n
is Avantage is (approximately) inversely proportional to the number of digits in n displaystyle n