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BINOMIAL COEFFICIENTS, CATALAN NUMBERS AND LUCAS QUOTIENTS
 SCI. CHINA MATH. 53(2010), IN PRESS.
, 2010
"... Let p be an odd prime and let a,m ∈ Z with a> 0 and p ∤ m. In this paper we determine ∑p a −1 ( 2k k=0 /mk mod p2 for d = 0,1; for k+d example, p a −1 k=0 ..."
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Cited by 36 (35 self)
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Let p be an odd prime and let a,m ∈ Z with a> 0 and p ∤ m. In this paper we determine ∑p a −1 ( 2k k=0 /mk mod p2 for d = 0,1; for k+d example, p a −1 k=0
Integrals of the Ising class
, 2006
"... From an experimentalmathematical perspective we analyze “Isingclass” integrals. These are structurally related ndimensional integrals we call Cn, Dn, En, where Dn is a magnetic susceptibility integral central to the Ising theory of solidstate physics. We first analyze ..."
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Cited by 26 (19 self)
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From an experimentalmathematical perspective we analyze “Isingclass” integrals. These are structurally related ndimensional integrals we call Cn, Dn, En, where Dn is a magnetic susceptibility integral central to the Ising theory of solidstate physics. We first analyze
Primality testing: variations on a theme of lucas
 Congressus Numerantium
"... Abstract. This survey traces an idea of Édouard Lucas that is a common element in various primality tests. These tests include those based on Fermat’s little theorem, elliptic curves, Lucas sequences, and polynomials over finite fields, including the recent test of Agrawal, Kayal, and Saxena. The Lu ..."
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Cited by 2 (0 self)
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Abstract. This survey traces an idea of Édouard Lucas that is a common element in various primality tests. These tests include those based on Fermat’s little theorem, elliptic curves, Lucas sequences, and polynomials over finite fields, including the recent test of Agrawal, Kayal, and Saxena. The Lucas idea may be summed up as follows: build up a group so large that n must be prime.
PRIME CHAINS AND PRATT TREES
"... ABSTRACT. 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 wit ..."
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Cited by 1 (0 self)
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ABSTRACT. 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. 1.
CONJECTURES INVOLVING ARITHMETICAL SEQUENCES
"... Abstract. We pose thirty conjectures on arithmetical sequences, most of which are about monotonicity of sequences of the form ( n √ an)n�1 or the form ( n+1 √ an+1 / n √ an)n�1, where (an)n�1 is a numbertheoretic or combinatorial sequence of positive integers. This material might stimulate further ..."
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Cited by 1 (1 self)
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Abstract. We pose thirty conjectures on arithmetical sequences, most of which are about monotonicity of sequences of the form ( n √ an)n�1 or the form ( n+1 √ an+1 / n √ an)n�1, where (an)n�1 is a numbertheoretic or combinatorial sequence of positive integers. This material might stimulate further research. 1.
TWO KINDS OF STRONG PSEUDOPRIMES UP TO 10 36
"... Abstract. Let n>1 be an odd composite integer. Write n − 1=2sd with d odd. If either bd ≡ 1modnor b2rd ≡−1modnfor some r =0, 1,...,s − 1, then we say that n isastrongpseudoprimetobaseb, or spsp(b) forshort. Define ψt to be the smallest strong pseudoprime to all the first t prime bases. If we know th ..."
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Abstract. Let n>1 be an odd composite integer. Write n − 1=2sd with d odd. If either bd ≡ 1modnor b2rd ≡−1modnfor some r =0, 1,...,s − 1, then we say that n isastrongpseudoprimetobaseb, or spsp(b) forshort. Define ψt to be the smallest strong pseudoprime to all the first t prime bases. If we know the exact value of ψt, we will have, for integers n<ψt, a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the ψt are known for 1 ≤ t ≤ 8. Conjectured values of ψ9,...,ψ12 were given by us in our previous papers (Math. Comp. 72 (2003), 2085–2097; 74 (2005), 1009–1024). The main purpose of this paper is to give exact values of ψ ′ t for 13 ≤ t ≤ 19; to give a lower bound of ψ ′ 20: ψ ′ 20> 1036; and to give reasons and numerical evidence of K2 and C3spsp’s < 1036 to support the following conjecture: ψt = ψ ′ t <ψ′ ′ t for any t ≥ 12, where ψ ′ t (resp. ψ′ ′ t) is the smallest K2 (resp. C3) strong pseudoprime to all the first t prime bases. For this purpose we describe procedures for computing and enumerating the two kinds of spsp’s < 1036 to the first 9 prime bases. The entire calculation took about 4000 hours on a PC Pentium IV/1.8GHz. (Recall that a K2spsp is an spsp of the form: n = pq with p, q primes and q − 1=2(p−1); and that a C3spsp is an spsp and a Carmichael number of the form: n = q1q2q3 with each prime factor qi ≡ 3mod4.) 1.
Algorithmic Number Theory MSRI Publications
"... Quadratic nonresidues 36 Chinese remainder theorem 57 ..."
unknown title
"... 1.1. Background. Lfunctions and modular forms underlie much of twentieth century number theory and are connected to the practical applications of number theory in cryptography. The fundamental importance of these functions in mathematics is supported by the fact that two of the seven Clay Mathemati ..."
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1.1. Background. Lfunctions and modular forms underlie much of twentieth century number theory and are connected to the practical applications of number theory in cryptography. The fundamental importance of these functions in mathematics is supported by the fact that two of the seven Clay Mathematics Million Dollar Millennium Problems [20] deal with properties of these functions, namely the