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PolynomialTime Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
 SIAM J. on Computing
, 1997
"... A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. ..."
Abstract

Cited by 1277 (4 self)
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A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration
Sum of prime factors in the prime factorization of an integer
 Int. Math. Forum
"... In memory of my sister Fedra Marina Jakimczuk (19702010) In this article we obtain some results on the sequence c(n), where c(n) is the sum of the prime factors in the prime factorization of n. Mathematics Subject Classification: 11A99, 11B99 ..."
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Cited by 1 (0 self)
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In memory of my sister Fedra Marina Jakimczuk (19702010) In this article we obtain some results on the sequence c(n), where c(n) is the sum of the prime factors in the prime factorization of n. Mathematics Subject Classification: 11A99, 11B99
ON THE PRIME FACTORIZATION OF BINOMIAL COEFFICIENTS
, 1978
"... For positive integers n and k, with n>2k, let (k = uv, where each prime factor of u is less than k, and each prime factor of v is at least equal to k. It is shown that u < v holds with just 12 exceptions, which are determined. If (k = UV, where each prime factor of U is at most equal to k, an ..."
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For positive integers n and k, with n>2k, let (k = uv, where each prime factor of u is less than k, and each prime factor of v is at least equal to k. It is shown that u < v holds with just 12 exceptions, which are determined. If (k = UV, where each prime factor of U is at most equal to k
Prime Factoring and The Complexity Of
"... Prime factorization is a mathematical problem with a long history. One of the oldest known methods of factoring is the Sieve of Eratosthenes. There have been numerous methods1 developed since the time of Eratosthenes. The Pollard Rho method2 is commonly accepted as the fastest publicly available fac ..."
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Prime factorization is a mathematical problem with a long history. One of the oldest known methods of factoring is the Sieve of Eratosthenes. There have been numerous methods1 developed since the time of Eratosthenes. The Pollard Rho method2 is commonly accepted as the fastest publicly available
Prime factors of dynamical sequences
, 2010
"... Let φ(t) ∈ Q(t) have degree d ≥ 2. For a given rational number x0, define xn+1 = φ(xn) for each n ≥ 0. If this sequence is not eventually periodic, then xn+1 − xn has a primitive prime factor for all sufficiently large n. This result provides a new proof of the infinitude of primes for each ratio ..."
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Cited by 11 (4 self)
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Let φ(t) ∈ Q(t) have degree d ≥ 2. For a given rational number x0, define xn+1 = φ(xn) for each n ≥ 0. If this sequence is not eventually periodic, then xn+1 − xn has a primitive prime factor for all sufficiently large n. This result provides a new proof of the infinitude of primes for each
ON THE PRIME FACTORS OF (n k)
"... A well known theorem of Sylvester and Schur (see [5]) states that for/7> 2k, the binomial coefficient l n.) always has a prime factor exceeding k. This can be considered as a generalization of the theorem of Chebysnev: There is always a prime between m and 2m. Set with ( n k) = un(k)vn(k) x un(k ..."
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Cited by 9 (2 self)
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A well known theorem of Sylvester and Schur (see [5]) states that for/7> 2k, the binomial coefficient l n.) always has a prime factor exceeding k. This can be considered as a generalization of the theorem of Chebysnev: There is always a prime between m and 2m. Set with ( n k) = un(k)vn(k) x un
Some congruences on prime factors of . . .
, 2002
"... • This paper is a contribution to the description of the odd prime factors of the class number of the number fields. • An example of the results obtained is: Let K/Q be an abelian extension with N = [K: Q]> 1, N odd. Let h(K) be the class number of K. Suppose that h(K)> 1. Let p be a prime div ..."
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• This paper is a contribution to the description of the odd prime factors of the class number of the number fields. • An example of the results obtained is: Let K/Q be an abelian extension with N = [K: Q]> 1, N odd. Let h(K) be the class number of K. Suppose that h(K)> 1. Let p be a prime
Factoring polynomials with rational coefficients
 MATH. ANN
, 1982
"... In this paper we present a polynomialtime algorithm to solve the following problem: given a nonzero polynomial fe Q[X] in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q[X]. It is well known that this is equivalent to factoring primitive polynomia ..."
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Cited by 961 (11 self)
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to be factored, n = deg(f) is the degree of f, and for a polynomial ~ a ~ i with real coefficients a i. i An outline of the algorithm is as follows. First we find, for a suitable small prime number p, a padic irreducible factor h of f, to a certain precision. This is done with Berlekamp's algorithm
ON THE GREATEST PRIME FACTOR OF ab+ 1
, 2008
"... Abstract. We prove that whenever A and B are dense enough subsets of {1,..., N}, there exist a ∈ A and b ∈ B such that the greatest prime factor of ab+ 1 is at least N1+A/(9N). 1. ..."
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Abstract. We prove that whenever A and B are dense enough subsets of {1,..., N}, there exist a ∈ A and b ∈ B such that the greatest prime factor of ab+ 1 is at least N1+A/(9N). 1.
A note on the primes in the prime factorization of an integer
 International Mathematical Forum
"... In this article we examine the primes that appear in the prime factorization of almost all numbers not exceeding x. Mathematics Subject Classification: 11A99 ..."
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Cited by 1 (1 self)
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In this article we examine the primes that appear in the prime factorization of almost all numbers not exceeding x. Mathematics Subject Classification: 11A99
Results 1  10
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