The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.

A set of m positive integers is called a Diophantine m-tuple 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.

Dedicated to the memory of Gian-Carlo 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 C-fractions, 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 η-function 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

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 three-valued Walsh spectrum can be transformed into 1-resilient functions. 1

A set of m positive integers is called a Diophantine m-tuple 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 9-tuple and that there are only finitely many Diophantine 8-tuples. 1

Diophantus noted that the rational numbers 1/16, 33/16, 17/4 and 105/16 have the following property: the product of any two of them increased by 1 is a square of a rational number. Let q be a rational number. A set of non-zero rationals {a1 , a2 , . . . , am} is called a rational Diophantine m-tuple with the property D(q) if a i a j + q is a square of a rational number for all 1 m.

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 p-integral 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

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

It is proved that if k and d are positive integers such that the product of any two distinct elements of the set {F_2k , F_2k+2 , F_2k+4 , d} increased by 1 is a perfect square, than d has to be 4F_2k+1 F_ 2k+2 F_2k+3. This is a generalization of the theorem of Baker and Davenport for k = 1.

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