In this paper, we describe an algorithm for solving systems of linear Diophantine equations based on a generalization of an algorithm for solving one equation due to Fortenbacher 3]. It can solve a system as a whole, or be used incrementally when the system is a sequential accumulation of several subsystems. The proof of termination of the algorithm is diicult, whereas the proofs of completenes...

We prove a law of the iterated logarithm for the Kolmogorov-Smirnov statistic, or equivalently, the discrepancy of sequences (nkω) mod 1. Here (nk) is a sequence of integers satisfying a subHadamard growth condition and such that linear Diophantine equations in the variables nk do not have too many solutions. The proof depends on a martingale embedding of the empirical process; the number-theor...

Attila Bérczes (University of Debrecen): On arithmetic properties of solutions of norm form equations. Abstract. Let α be an algebraic number of degree n and K := Q(α). Consider the norm form equation NK/Q(x0 + x1α+ x2α + . . .+ xn−1α) = b in x0, . . . , xn−1 ∈ Z. (1) Let H denote the solution set of (1). Arranging the elements of H in an |H| × n array H, one may ask at least two natural questi...

In 1900 Hubert asked for an algorithm to decide the solvability of all diophantine equations, P(x1, . . . , xv) = 0, where P is a polynomial with integer coefficients. In special cases of Hilbert's tenth problem, such algorithms are known. Siegel [7] gives an algorithm for all polynomials P(xx, . . . , xv) of degree < 2. From the work of A. Baker [1] we know that there is also a decision proced...

This is a report on the recent work by Claude Levesque and the author on families of Diophantine equations. This joint work started in 2010 in Rio, and this is still work in progress. The lecture in Lahore on March 11, 2013 was mainly devoted to a survey of results on Diophantine equations, with the last part dealing with some recent results. Here we describe the content of the recent joint pap...

1. Historical introduction. Many questions in number theory concern perfect powers, numbers of the form a b where a and b are rational integers with <7>1, 6>1. To mention a few: (a) Is it possible that for /zs>3 the sum of two 77th powers is an /7th power? (b) Is 8, 9 the only pair of perfect powers which differ by 1 ? (c) Is it possible that the product of consecutive integers, (x+l)(x + 2) .....

We establish a law of the iterated logarithm for the discrepancy of sequences (nkx) mod 1 where (nk) is a sequence of integers satisfying a sub-Hadamard growth condition and such that one and four-term Diophantine equations in the variables nk do not have too many solutions. The conditions are discussed, the probabilistic details of the proof are given elsewhere. As a corollary to our results, ...

Systems based on the splicing operation are computationally complete. Usually demonstrations of this are based on simulations of type-0 grammars. We propose a diierent way to reach this result by solving Diophantine equations using extended H system with permitting context. Completeness then follows from Matiyasevich's theorem stating that the class of Diophantine sets is identical to the class...

A significant portion of MacMahon’s famous book “Combinatory Analysis” is devoted to the development of “Partition Analysis” as a computational method for solving problems in connection with linear homogeneous diophantine inequalities and equations, respectively. Nevertheless, MacMahon’s ideas have not received due attention with the exception of work by Richard Stanley. A long range object of ...

The postage stamp problem and its variations are classical problems in elementary number theory. This problem involves finding the largest value that cannot be expressed as a nonnegative linear combination of two relatively prime integers m and p. It turns out that this value can be defined as a function of m and p. An original proof of this fact is the first major goal of this paper. Moreover,...