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Irreducible radical extensions and Eulerfunction chains, Combinatorial number theory
 Proc. Integers Conf. 2005), 351361, de Gruyter
, 2007
"... For Ron Graham on his 70th birthday We discuss the smallest algebraic number eld which contains the nth roots of unity and which may be reached from the rational eld Q by a sequence of irreducible, radical, Galois extensions. The degree D(n) of this eld over Q is '(m), where m is the smallest m ..."
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For Ron Graham on his 70th birthday We discuss the smallest algebraic number eld which contains the nth roots of unity and which may be reached from the rational eld Q by a sequence of irreducible, radical, Galois extensions. The degree D(n) of this eld over Q is '(m), where m is the smallest multiple of n divisible by each prime factor of '(m). The prime factors of m=n are precisely the primes not dividing n but which do divide some number in the \Euler chain " '(n); '('(n)); : : :. For each xed k, we show that D(n)> nk on a set of asymptotic density 1.
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.
Open Problems on Exponential and Character Sums
, 2010
"... This is a collection of mostly unrelated open questions, at various levels of difficulty, related to exponential and multiplicative character sums. One may certainly notice a large proportion of selfreferences in the bibliography. By no means should this be considered as an indication of anything e ..."
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Cited by 2 (0 self)
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This is a collection of mostly unrelated open questions, at various levels of difficulty, related to exponential and multiplicative character sums. One may certainly notice a large proportion of selfreferences in the bibliography. By no means should this be considered as an indication of anything else than
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.
On shifted primes and balanced primes
 Int. J. Number Theory
"... The asymptotic order of the number of primes, which are such that the shift by a fixed integer is a number supported by a given set of primes times a coprime squarefree number, is determined. The order is also determined when the shift and its negative have this same shape. In each case the order is ..."
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The asymptotic order of the number of primes, which are such that the shift by a fixed integer is a number supported by a given set of primes times a coprime squarefree number, is determined. The order is also determined when the shift and its negative have this same shape. In each case the order is dependent only on the set of primes and the squarefree core of the shift. 1
Divisibility, Smoothness and Cryptographic Applications
, 2008
"... This paper deals with products of moderatesize primes, familiarly known as smooth numbers. Smooth numbers play an crucial role in information theory, signal processing and cryptography. We present various properties of smooth numbers relating to their enumeration, distribution and occurrence in var ..."
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This paper deals with products of moderatesize primes, familiarly known as smooth numbers. Smooth numbers play an crucial role in information theory, signal processing and cryptography. We present various properties of smooth numbers relating to their enumeration, distribution and occurrence in various integer sequences. We then turn our attention to cryptographic applications in which smooth numbers play a pivotal role. 1 1
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7(2) (2007), #A25 IRREDUCIBLE RADICAL EXTENSIONS AND EULERFUNCTION CHAINS
"... We discuss the smallest algebraic number field which contains the nth roots of unity and which may be reached from the rational field Q by a sequence of irreducible, radical, Galois extensions. The degree D(n) of this field over Q is ϕ(m), where m is the smallest multiple of n divisible by each prim ..."
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We discuss the smallest algebraic number field which contains the nth roots of unity and which may be reached from the rational field Q by a sequence of irreducible, radical, Galois extensions. The degree D(n) of this field over Q is ϕ(m), where m is the smallest multiple of n divisible by each prime factor of ϕ(m). The prime factors of m/n are precisely the primes not dividing n but which do divide some number in the “Euler chain ” ϕ(n), ϕ(ϕ(n)),.... For each fixed k, we show that D(n)> n k on a set of asymptotic density 1. 1.