항목
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On the Number of Primes Less Than a Given Magnitude 주어진 수보다 작은 소수의 개수에 관하여, 与えられた数より小さ..the Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin.This paper studies the prime-counting function using analytic methods. Although it is the only paper Riemann ever published on number theory, it contains ideas which influenced thousands of researchers during the...출처 영어 위키백과
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Primorial 소수 계승, 素数階乗to Harvey Dubner, draws an analogy to primes the same way the name "factorial" relates to factors. Definition for Primorial numbers For the nth prime number pn the primorial pn# is defined as the product of the first n primes:p_n\# = \prod_{k=1}^n p_k where pk is the kth prime number.For instance...출처 영어 위키백과
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Formula for primes 소수 공식, 素数公式a "formula" can and cannot be. It is known that no non-constant polynomial function P(n) with integer coefficients exists that evaluates to a prime number for all integers n. The proof is as follows: Suppose such a polynomial existed. Then P(1) would evaluate to a prime p, so P(1) \equiv 0 \pmod...출처 영어 위키백과
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Miller–Rabin primality test 밀러-라빈 소수판별법, ミラー-ラビン素..or a^{2^r\cdot d} \equiv -1\pmod{n} for some 0 ≤ r ≤ s − 1.To show that one of these must be true, recall Fermat's little theorem, that for a prime number n: a^{n-1} \equiv 1\pmod{n}. By the lemma above, if we keep taking square roots of an−1, we will get either 1 or −1. If we get −1 then...출처 영어 위키백과
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Divergence of the sum of the reciprocals of the primes 소수의 역수의 합의 발산성, 素..1}} \,\geq\, \frac{1}{p_i - 1}. Hence \sum_p \frac{1}{p} diverges. The following proof by contradiction is due to Paul Erdős.Let pi denote the ith prime number. Assume that the sum of the reciprocals of the primes converges; i.e., \sum_{i=1}^\infty {1\over p_i} < \infty Then there exists a...출처 영어 위키백과
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Lucas–Lehmer primality test 뤼카-레머 소수판별법, 卢卡斯-莱默检验法AKS primality test, requires Õ(n6) bit operations in its best known variant and is dramatically slower in practice. Examples The Mersenne number M3 = 7 is prime. The Lucas–Lehmer test verifies this as follows. Initially s is set to 4 and then is updated 3−2 = 1 time: s ← ((4 × 4) − 2) mod 7...출처 영어 위키백과
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AKS primality test AKS 소수판별법, AKS素数判定法at the Indian Institute of Technology Kanpur, on August 6, 2002, in a paper titled "PRIMES is in P". The algorithm determines whether a number is prime or composite within polynomial time. The authors received the 2006 Gödel Prize and the 2006 Fulkerson Prize for this work. AKS is the first...출처 영어 위키백과
