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The Lucas–Pratt primality tree
 Math. Comp
"... Abstract. In 1876, E. Lucas showed that a quick proof of primality for a prime p could be attained through the prime factorization of p − 1 and a primitive root for p. V. Pratt’s proof that PRIMES is in NP, done via Lucas’s theorem, showed that a certificate of primality for a prime p could be obtai ..."
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Abstract. In 1876, E. Lucas showed that a quick proof of primality for a prime p could be attained through the prime factorization of p − 1 and a primitive root for p. V. Pratt’s proof that PRIMES is in NP, done via Lucas’s theorem, showed that a certificate of primality for a prime p could be obtained in O(log 2 p) modular multiplications with integers at most p. We show that for all constants C ∈ R, the number of modular multiplications necessary to obtain this certificate is greater than C log p for a set of primes p with relative asymptotic density 1. 1.
COMMON VALUES OF THE ARITHMETIC FUNCTIONS φ AND σ
"... ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sumofdivisors function. This proves a 50year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each ..."
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ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sumofdivisors function. This proves a 50year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each have more than n c solutions, for some c> 0. The proofs rely on the recent work of the first two authors and Konyagin on the distribution of primes p for which a given prime divides some iterate of φ at p, and on a result of HeathBrown connecting the possible existence of Siegel zeros with the distribution of twin primes. 1.
PRIME CHAINS AND PRATT TREES
, 2009
"... We study the distribution of prime chains, which are sequences p1,..., pk of primes for which pj+1 ≡ 1 (mod pj) for each j. We first give conditional upper bounds on the length of Cunningham chains, chains with pj+1 = 2pj +1 for each j. We give estimates for P (x), the number of chains with pk � x ..."
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We study the distribution of prime chains, which are sequences p1,..., pk of primes for which pj+1 ≡ 1 (mod pj) for each j. We first give conditional upper bounds on the length of Cunningham chains, chains with pj+1 = 2pj +1 for each j. We give estimates for P (x), the number of chains with pk � x (k variable), and P (x; p), the number of chains with p1 = p and pk � px. The majority of the paper concerns the distribution of H(p), the length of the longest chain with pk = p, which is also the height of the Pratt tree for p. We show H(p) � c log log p and H(p) � (log p) 1−c′ for almost all p, with c, c ′ explicit positive constants. We can take, for any ε> 0, c = e − ε assuming the ElliottHalberstam conjecture. A stochastic model of the Pratt tree is introduced and analyzed. The model suggests that for most p � x, H(p) stays very close to e log log x.
SIEVING VERY THIN SETS OF PRIMES, AND PRATT TREES WITH MISSING PRIMES
"... ABSTRACT. Suppose P is a set of primes, such that for every p ∈ P, every prime factor of p − 1 is also in P. We apply a new sieve method to show that either P contains all of the primes or the counting function of P is O(x 1−c) for some c> 0, where c depends only on the smallest prime not in P. O ..."
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ABSTRACT. Suppose P is a set of primes, such that for every p ∈ P, every prime factor of p − 1 is also in P. We apply a new sieve method to show that either P contains all of the primes or the counting function of P is O(x 1−c) for some c> 0, where c depends only on the smallest prime not in P. Our proof makes use of results connected with Artin’s primitive root conjecture. 1
Previous Up Next Article Citations From References: 5 From Reviews: 0
"... Spectral factorization of biinfinite multiindex block Toeplitz matrices. (English summary) Special issue on structured and infinite systems of linear equations. Linear Algebra Appl. 343/344 (2002), 355–380. Block lowerdiagonalupper LDU and Cholesky LL ∗ factorizations are studied for weighted Wi ..."
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Spectral factorization of biinfinite multiindex block Toeplitz matrices. (English summary) Special issue on structured and infinite systems of linear equations. Linear Algebra Appl. 343/344 (2002), 355–380. Block lowerdiagonalupper LDU and Cholesky LL ∗ factorizations are studied for weighted Wiener classes of biinfinite block Toeplitz matrices A = (Ai−j) i,j∈Zd, where Aj are complex k × k matrices, and ∑ i∈Zd βi‖Ai ‖ < ∞, for a fixed sequence of weights {βi} i∈Zd subject to 1 ≤ βi+j ≤ βiβj. A further generalization, also studied in the paper under review, consists of considering the biinfinite matrices A with the additional requirement that Ai = 0 for i ̸ ∈ J, where J is a fixed subgroup of Zd. The factorizations are considered with respect to a total order in Zd. One result: If A is positive definite, satisfies the additional requirement, and the symbol of A is invertible on the maximal ideal space, then A has a Cholesky factorization, where the factor L and its inverse also satisfy the additional requirement and belong to the Wiener class with the trivial weights βi ≡ 1; in the scalar case k = 1 the factor L in fact belongs to the weighted Wiener class. Convergence of projection methods for computing the Cholesky factorization is proved and illustrated by an example. A comparison is made with Krein’s method, which is based on the additive decomposition of the logarithm of A (when applicable).
Journal of Integer Sequences, Vol. 16 (2013), Article 13.5.5 On the Middle Prime Factor of an Integer
"... Given an integer n ≥ 2, let pm(n) denote the middle prime factor of n. We obtain an estimate for the sum of the reciprocals of pm(n) for n ≤ x. 1 ..."
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Given an integer n ≥ 2, let pm(n) denote the middle prime factor of n. We obtain an estimate for the sum of the reciprocals of pm(n) for n ≤ x. 1
On Positive Integers n with a Certain Divisibility Property
"... In this paper, we study the positive integers n having at least two distinct prime factors such that the sum of the prime factors of n divides 2 n−1 − 1. 1 ..."
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In this paper, we study the positive integers n having at least two distinct prime factors such that the sum of the prime factors of n divides 2 n−1 − 1. 1