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18
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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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 polynomials feZ[X] into irreducible factors in Z[X]. Here we call f ~ Z[X] primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. [8]. Its running time, measured in bit operations, is O(nl2+n9(log[fD3). Here f~Tl[X] is the polynomial 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 for factoring polynomials over small finite fields, combined with Hensel's lemma. Next we look for the irreducible factor h o of f in
Computing π(x): An analytic method
 J. Algorithms
, 1987
"... The problem of computing π(x), the number of primes p ≤ x, is a very old one. The sieve of Eratosthenes, in its usual form, finds all primes p ≤ x in O(x log log x) steps, where we consider for our model of computation a random access machine with about ..."
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Cited by 8 (2 self)
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The problem of computing π(x), the number of primes p ≤ x, is a very old one. The sieve of Eratosthenes, in its usual form, finds all primes p ≤ x in O(x log log x) steps, where we consider for our model of computation a random access machine with about
Approximating the number of integers without large prime factors
 Mathematics of Computation
, 2004
"... Abstract. Ψ(x, y) denotes the number of positive integers ≤ x and free of prime factors>y. Hildebrand and Tenenbaum gave a smooth approximation formula for Ψ(x, y) in the range (log x) 1+ɛ <y ≤ x,whereɛ is a fixed positive number ≤ 1/2. In this paper, by modifying their approximation formula, ..."
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Abstract. Ψ(x, y) denotes the number of positive integers ≤ x and free of prime factors>y. Hildebrand and Tenenbaum gave a smooth approximation formula for Ψ(x, y) in the range (log x) 1+ɛ <y ≤ x,whereɛ is a fixed positive number ≤ 1/2. In this paper, by modifying their approximation formula, we provide a fast algorithm to approximate Ψ(x, y). The computational complexity of this algorithm is O ( � (log x)(log y)). We give numerical results which show that this algorithm provides accurate estimates for Ψ(x, y) andisfaster than conventional methods such as algorithms exploiting Dickman’s function. 1.
Iterated absolute values of differences of consecutive primes
 Mathematics of Computation
, 1993
"... Abstract. Let dç,(n) = p „ , the nth prime, for n> 1, and let dk+x(n) = \dk(n) dk(n + 1)  for k> 0, n> 1. A wellknown conjecture, usually ascribed to Gilbreath but actually due to Proth in the 19th century, says that dk(\) = 1 for all k> 1. This paper reports on a computation that v ..."
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Abstract. Let dç,(n) = p „ , the nth prime, for n> 1, and let dk+x(n) = \dk(n) dk(n + 1)  for k> 0, n> 1. A wellknown conjecture, usually ascribed to Gilbreath but actually due to Proth in the 19th century, says that dk(\) = 1 for all k> 1. This paper reports on a computation that verified this conjecture for k < tt(1013) » 3 x 10 ". It also discusses the evidence and the heuristics about this conjecture. It is very likely that similar conjectures are also valid for many other integer sequences. 1.
A Wieferich Prime Search up to 6.7 × 10^15
 JOURNAL OF INTEGER SEQUENCES, VOL. 14 (2011), ARTICLE 11.9.2
, 2011
"... A Wieferich prime is a prime p such that 2 p−1 ≡ 1 (mod p 2). Despite several intensive searches, only two Wieferich primes are known: p = 1093 and p = 3511. This paper describes a new search algorithm for Wieferich primes using doubleprecision Montgomery arithmetic and a memoryless sieve, which ru ..."
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A Wieferich prime is a prime p such that 2 p−1 ≡ 1 (mod p 2). Despite several intensive searches, only two Wieferich primes are known: p = 1093 and p = 3511. This paper describes a new search algorithm for Wieferich primes using doubleprecision Montgomery arithmetic and a memoryless sieve, which runs significantly faster than previously published algorithms, allowing us to report that there are no other Wieferich primes p < 6.7 × 10 15. Furthermore, our method allowed for the efficent collection of statistical data on Fermat quotients, leading to a strong empirical confirmation of a conjecture of Crandall, Dilcher, and Pomerance. Our methods proved flexible enough to search for new solutions of a p−1 ≡ 1 (mod p 2) for other small values of a, and to extend the search for FibonacciWieferich primes. We conclude, among other things, that there are no FibonacciWieferich primes less than p < 9.7 × 10 14.
The Pseudosquares Prime Sieve
"... Abstract. We present the pseudosquares prime sieve, which finds all primes up to n. Define p to be the smallest prime such that the pseudosquare Lp>n/(π(p)(log n) 2); here π(x) is the prime counting function. Our algorithm requires only O(π(p)n) arithmetic operations and O(π(p)logn) space. It use ..."
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Abstract. We present the pseudosquares prime sieve, which finds all primes up to n. Define p to be the smallest prime such that the pseudosquare Lp>n/(π(p)(log n) 2); here π(x) is the prime counting function. Our algorithm requires only O(π(p)n) arithmetic operations and O(π(p)logn) space. It uses the pseudosquares primality test of Lukes, Patterson, and Williams. Under the assumption of the Extended Riemann Hypothesis, we have p ≤ 2(log n) 2, but it is conjectured that p ∼ 1 log nlog log n. Thus, log2 the conjectured complexity of our prime sieve is O(n log n) arithmetic operations in O((log n) 2) space. The primes generated by our algorithm are proven prime unconditionally. The best current unconditional bound known is p ≤ n 1/(4√e−ɛ) 1.132, implying a running time of roughly n using roughly n 0.132 space. Existing prime sieves are generally faster but take much more space, greatly limiting their range (O(n / log log n)operationswithn 1/3+ɛ space, or O(n) operationswithn 1/4 conjectured space). Our algorithm found all 13284 primes in the interval [10 33,10 33 +10 6] in about 4 minutes on a1.3GHzPentiumIV. We also present an algorithm to find all pseudosquares Lp up to n in sublinear time using very little space. Our innovation here is a new, spaceefficient implementation of the wheel datastructure. 1
Improved Incremental Prime Number Sieves
 Cornell University
, 1994
"... . An algorithm due to Bengalloun that continuously enumerates the primes is adapted to give the first prime number sieve that is simultaneously sublinear, additive, and smoothly incremental:  it employs only \Theta(n= log log n) additions of numbers of size O(n) to enumerate the primes up to n, ..."
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. An algorithm due to Bengalloun that continuously enumerates the primes is adapted to give the first prime number sieve that is simultaneously sublinear, additive, and smoothly incremental:  it employs only \Theta(n= log log n) additions of numbers of size O(n) to enumerate the primes up to n, equalling the performance of the fastest known algorithms for fixed n;  the transition from n to n + 1 takes only O(1) additions of numbers of size O(n). (On average, of course, O(1) such additions increase the limit up to which all primes are known from n to n + \Theta(log log n)). 1 Introduction A socalled "formula" for the i'th prime has been a longlived concern, if not quite the Holy Grail, of Elementary Number Theory. This concern seems poorly motivated, as evidenced by the extraordinary freakshow of solutions proffered over the ages. The natural setting is Algorithmic Number Theory, and what is desired is much better cast as an algorithm to compute the i'th prime. Given that app...
A fully distributed prime numbers generation using the wheel sieve, Parallel and Distributed Computing and networks
, 2005
"... This article presents a new distributed approach for generating all prime numbers up to a given limit. From Eratosthenes, who elaborated the first prime sieve (more than 2000 years ago), to the advances of the parallel computers (which have permitted to reach large limits or to obtain the previou ..."
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This article presents a new distributed approach for generating all prime numbers up to a given limit. From Eratosthenes, who elaborated the first prime sieve (more than 2000 years ago), to the advances of the parallel computers (which have permitted to reach large limits or to obtain the previous results in a shorter time), prime numbers generation still represents an attractive domain of research. Nowadays, prime numbers play a central role in cryptography and their interest has been increased by the very recent proof that primality testing is in P. In this work, we propose a new distributed algorithm which generates all prime numbers in a given finite interval [2,..., n], based on the wheel sieve. As far as we know, this paper designs the first fully distributed wheel sieve algorithm. KEY WORDS Distributed algorithms, prime numbers generation, wheel sieve, broadcast and leader election. 1
A Distributed Prime Sieving Algorithm based on Scheduling by Multiple Edge Reversal
 In: 4th International Symposium on Parallel and Distributed Computing, 2005, Lille. Proceedings of the 4th International Symposium on Parallel and Distributed Computing. Los Alamitos
, 2005
"... Abstract — In this article, we propose a fully distributed algorithm for finding all primes in an given interval [2..n] (or (L, R), more generally), based on the SMER — Scheduling by Multiple Edge Reversal — multigraph dynamics. Given a multigraph M of arbitrary topology, having N nodes, the SMERdr ..."
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Abstract — In this article, we propose a fully distributed algorithm for finding all primes in an given interval [2..n] (or (L, R), more generally), based on the SMER — Scheduling by Multiple Edge Reversal — multigraph dynamics. Given a multigraph M of arbitrary topology, having N nodes, the SMERdriven system is defined by the number of directed edges (arcs) between any two nodes of M, and by the global period length of all “arc reversals ” in M. In the domain of prime numbers generation, such a graph method shows quite elegant, and it also yields a totally new kind of distributed prime sieving algorithms of an entirely original design. The maximum number of steps required by the algorithm is at most n + √ n. Although far beyond the O(n / log log n) steps required by the improved sequential “wheel sieve ” algorithms, our SMERbased algorithm is fully distributed and of linear (step) complexity. The message complexity of the algorithm is at most n∆N + √ n∆N, where ∆N denotes the maximum “multidegree ” of the arbitrary multigraph M, and the space required per process is linear.
Trading Time for Space in Prime Number Sieves
 Proceedings of the Third International Algorithmic Number Theory Symposium (ANTS III
, 1998
"... . A prime number sieve is an algorithm that finds the primes up to a bound n. We present four new prime number sieves. Each of these sieves gives new space complexity bounds for certain ranges of running times. In particular, we give a linear time sieve that uses only O( p n=(log log n) 2 ) bit ..."
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. A prime number sieve is an algorithm that finds the primes up to a bound n. We present four new prime number sieves. Each of these sieves gives new space complexity bounds for certain ranges of running times. In particular, we give a linear time sieve that uses only O( p n=(log log n) 2 ) bits of space, an O l (n= log log n) time sieve that uses O(n=((log n) l log log n)) bits of space, where l ? 1 is constant, and two superlinear time sieves that use very little space. 1 Introduction A prime number sieve is an algorithm that finds all prime numbers up to a bound n. In this paper we present four new prime number sieves, three of which accept a parameter to control their use of time versus space. The fastest known prime number sieve is the dynamic wheel sieve of Pritchard [11], which uses O(n= log log n) arithmetic operations and O(n= log log n) bits of space. Dunten, Jones, and Sorenson [6] gave an algorithm with the same asymptotic running time, while using only O(n=(log ...