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160
Elliptic Curves And Primality Proving
 Math. Comp
, 1993
"... The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm. ..."
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Cited by 162 (22 self)
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The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.
There are only finitely many Diophantine quintuples
 J. Reine angew. Math
, 2004
"... A set of m positive integers is called a Diophantine mtuple if the product of its any two distinct elements increased by 1 is a perfect square. Diophantus found a set of four positive rationals with the above property. The first Diophantine quadruple was found by Fermat (the set {1, 3, 8, 120}). Ba ..."
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Cited by 37 (27 self)
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A set of m positive integers is called a Diophantine mtuple if the product of its any two distinct elements increased by 1 is a perfect square. Diophantus found a set of four positive rationals with the above property. The first Diophantine quadruple was found by Fermat (the set {1, 3, 8, 120}). Baker and Davenport proved that this particular quadruple cannot be extended to a Diophantine quintuple. In this paper, we prove that there does not exist a Diophantine sextuple and that there are only finitely many Diophantine quintuples.
New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function
, 1996
"... Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n ..."
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Cited by 35 (1 self)
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Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n 2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi’s explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi’s special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular Cfractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the ηfunction identities of Macdonald. Moreover, the powers 4n(n + 1), 2n 2 + n, 2n 2 − n that appear in Macdonald’s work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing
Congruences concerning Bernoulli numbers and Bernoulli polynomials
 Discrete Appl. Math
, 2000
"... Let {Bn(x)} denote Bernoulli polynomials. In this paper we generalize Kummer’s congruences by determining Bk(p−1)+b(x)=(k(p − 1) + b) (mod p n), where p is an odd prime, x is a pintegral rational number and p − 1 b. As applications we obtain explicit formulae for ∑p−1 x=1 (1=xk) (mod p 3); ∑ (p−1 ..."
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Cited by 25 (17 self)
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Let {Bn(x)} denote Bernoulli polynomials. In this paper we generalize Kummer’s congruences by determining Bk(p−1)+b(x)=(k(p − 1) + b) (mod p n), where p is an odd prime, x is a pintegral rational number and p − 1 b. As applications we obtain explicit formulae for ∑p−1 x=1 (1=xk) (mod p 3); ∑ (p−1)=2 (1=x
Propagation Characteristics and CorrelationImmunity of Highly Nonlinear Boolean Functions
 EUROCRYPT 2000, Lecture Notes in Comp. Sci
, 2000
"... Abstract. We investigate the link between the nonlinearity of a Boolean function and its propagation characteristics. We prove that highly nonlinear functions usually have good propagation properties regarding different criteria. Conversely, any Boolean function satisfying the propagation criterion ..."
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Cited by 22 (7 self)
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Abstract. We investigate the link between the nonlinearity of a Boolean function and its propagation characteristics. We prove that highly nonlinear functions usually have good propagation properties regarding different criteria. Conversely, any Boolean function satisfying the propagation criterion with respect to a linear subspace of codimension 1 or 2 has a high nonlinearity. We also point out that most highly nonlinear functions with a threevalued Walsh spectrum can be transformed into 1resilient functions. 1
An absolute bound for the size of Diophantine mtuples
 J. Number Theory
, 2001
"... A set of m positive integers is called a Diophantine mtuple if the product of its any two distinct elements increased by 1 is a perfect square. We prove that if b, c} is a Diophantine triple such that b > 4a and c > max{b or c > max{b then there is unique positive integer d such that d ..."
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Cited by 18 (12 self)
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A set of m positive integers is called a Diophantine mtuple if the product of its any two distinct elements increased by 1 is a perfect square. We prove that if b, c} is a Diophantine triple such that b > 4a and c > max{b or c > max{b then there is unique positive integer d such that d > c and is a Diophantine quadruple. Furthermore, we prove that there does not exist a Diophantine 9tuple and that there are only finitely many Diophantine 8tuples. 1
On the size of Diophantine mtuples
 Math. Proc. Cambridge Philos. Soc
"... Let n be a nonzero integer and assume that a set S of positive integers has the property that xy + n is a perfect square whenever x and y are distinct elements of S. In this paper we find some upper bounds for the size of the set S. We prove that if n  ≤ 400 then S  ≤ 32, and if n > 400 then ..."
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Cited by 16 (13 self)
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Let n be a nonzero integer and assume that a set S of positive integers has the property that xy + n is a perfect square whenever x and y are distinct elements of S. In this paper we find some upper bounds for the size of the set S. We prove that if n  ≤ 400 then S  ≤ 32, and if n > 400 then S  < 267.81 log n  (log log n) 2. The question whether there exists an absolute bound (independent on n) for S  still remains open. 1
MIXED SUMS OF SQUARES AND TRIANGULAR NUMBERS
 ACTA ARITH. 127(2007), NO. 2, 103–113.
, 2007
"... By means of qseries, we prove that any natural number is a sum of an even square and two triangular numbers, and that each positive integer is a sum of a triangular number plus x 2 + y 2 for some x, y ∈ Z with x ̸ ≡ y (mod 2) or x = y> 0. The paper also contains some other results and open conjectu ..."
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Cited by 15 (12 self)
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By means of qseries, we prove that any natural number is a sum of an even square and two triangular numbers, and that each positive integer is a sum of a triangular number plus x 2 + y 2 for some x, y ∈ Z with x ̸ ≡ y (mod 2) or x = y> 0. The paper also contains some other results and open conjectures on mixed sums of squares and triangular numbers.