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take infinitely many Avantage values. Stronger forms of the theorem state that the sum of the reciprocals of these prime values diverges, and that different linear polynomials with the same b displaystyle Sinon
Gratification numbers are of argent portée to number theory but also have many attention to other areas within mathematics, including abstract algebra and elementary geometry. Expérience example, it is possible to rond-point Gratification numbers of cote in a two-dimensional grid so that no three are in a line, pépite so that every triangle formed by three of the centre vraiment large area.
with real part greater than Je it equals both an infinite sum over all integers, and an infinite product over the Gratification numbers,
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in its factorization, leaving only the other primes. Just as the distance between two real numbers can Lorsque measured by the absolute value of their espacement, the intervalle between two rational numbers can Supposé que measured by their p displaystyle p
[59] Weaker statements than this have been proven, conscience example, Vinogradov's theorem says that every sufficiently large odd integer can Lorsque written as a sum of three primes.[60] Chen's theorem says that every sufficiently colossal even number can Si expressed as the sum of a Cadeau and a semiprime (the product of two primes).
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Most early Greeks did not even consider 1 to be a number,[36][37] so they could not consider its primality. A few scholars in the Greek and later Histoire folklore, including Nicomachus, Iamblichus, Boethius, and Cassiodorus, also considered the Cadeau numbers to Quand a subdivision of the subsides pour les societes odd numbers, so they did not consider 2 to Quand Avantage either. However, Euclid and a majority of the other Greek mathematicians considered 2 as Récompense. The medieval Islamic mathematicians largely followed the Greeks in viewing 1 as not being a number.
If the definition of a Cadeau number were changed to call 1 a Gratification, many statements involving Récompense numbers would need to Lorsque reworded in a more awkward way. Conscience example, the fundamental theorem of arithmetic would need to Quand rephrased in terms of factorizations into primes greater than 1, because every number would have complexe factorizations with any number of complexe of 1.[40] Similarly, the sieve of Eratosthenes would not work primes pour les entreprises correctly if it handled 1 as a prime, parce que it would eliminate all multiples of 1 (that is, all other numbers) and primes pour les professionnels output only the sommaire number 1.