Abstract. Let K be either the real, complex, or quaternion number system and let O(K) be the corresponding integers. Let x = (x 1 , . . . , xn) be a vector in K n . The vector x has an integer relation if there exists a vector m = (m 1 , . . . , mn) ∈ O(K) n , m = 0, such that m 1 x 1 + m 2 x 2 + . . . + mnxn = 0. In this paper we define the parameterized integer relation construction algorithm PSLQ(τ ), where the parameter τ can be freely chosen in a certain interval.Beginning with an arbitrary vector x = (x 1 , . . . , xn) ∈ K n , iterations of PSLQ(τ ) will produce lower bounds on the norm of any possible relation for x. Thus PSLQ(τ) can be used to prove that there are no relations for x of norm less than a given size. Let Mx be the smallest norm of any relation for x. For the real and complex case and each fixed parameter τ in a certain interval, we prove that PSLQ(τ ) constructs a relation in less than O(n 3 + n 2 log Mx) iterations.

ABSTRACT. A construction using integral matrices with determinant ± 1 is given which has as corollaries generalizations of classical theorems of Dirichlet and Kronecker. This construction yields a geometrically convergent algorithm successfully generalizing the Euclidean algorithm to finite sets of real numbers. Applied to such a set this algorithm terminates if and only if the set is integrally linearly dependent and the algorithm gives absolute simultaneous integral approximations if and only if the set is integrally linearly independent. This development applies to complex numbers, can be used to give proofs of irreducibility of polynomials and yields effective lower bounds on heights of integral relations.Let Z = rational integers, R = real numbers, Z" = lattice points C R" as row vectors, GL W (Z) C GL W (R) are n by n matrices with entries and invertible determinants in Z C R resp. For M = any matrix or vector, M f = transpose, row,.M = ith row, coljM = ;th column, height (M) = max absolute values of entries of M. The entries ofxGR" are Z-linearly dependent iff there exists 0 =£ m G Z n such that xm f = 0, m = Z-relation for x For 0 &x E R n

Abstract.Let i = (11,12, • • •, xn) be a vector of real numbers, x is said to possess an integer relation if there exist integers a,-not all zero such that ain + a2X2 + ■ ■ ■ + anxn = 0. Beginning ten years ago, algorithms were discovered by one of us which, for any n, are guaranteed to either find a relation if one exists, or else establish bounds within which no relation can exist. One of those algorithms has been employed to study whether or not certain fundamental mathematical constants satisfy simple algebraic polynomials.Recently, one of us discovered a new relation-finding algorithm that is much more efficient, both in terms of run time and numerical precision. This algorithm has now been implemented on high-speed computers, using multiprecision arithmetic. With the help of these programs, several of the previous numerical results on mathematical constants have been extended, and other possible relationships between certain constants have been studied. This paper describes this new algorithm, summarizes the numerical results, and discusses other possible applications.In particular, it is established that none of the following constants satisfies a simple, low-degree polynomial: 7 (Euler's constant), log'y, log7r, pi (the imaginary part of the first zero of Riemann's zeta function), logpi, f (3) (Riemann's zeta function evaluated at 3), and logç(3). Several classes of possible additive and multiplicative relationships between these and related constants are ruled out. Results are also cited for Feigenbaum's constant, derived from the theory of chaos, and two constants of fundamental physics, derived from experiment.

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