The wish to determine the complete set of rational integral solutions of (1) was expressed by Diaconis and Graham in [2, p. 328]. Apparently, the solutions to this diophantine problem correspond to values of keN for which the Radon transform based on the set of all xeZ\ with exactly four ones is not invertible. We thank Hendrik Lenstra who communicated the problem to Jaap Top, to whom we are eq...

Let f(X1, . . . , Xm) be a quadratic form in m variables X1, . . . , Xm with integer coefficients. Then it is well-known that the Diophantine equation f(X1, . . . , Xm) = 0 has a nontrivial solution in integers if and only if the equation has a nontrivial solution in real numbers and the congruence f(X1, . . . , Xm) ≡ 0 (mod N) has a nontrivial solution for every integer N > 1. Such a principle...

There exist many results about the Diophantine equation (qn − 1)/(q − 1) = ym, where m ≥ 2 and n ≥ 3. In this paper, we suppose that m = 1, n is an odd integer and q a power of a prime number. Also let y be an integer such that the number of prime divisors of y − 1 is less than or equal to 3. Then we solve completely the Diophantine equation (qn − 1)/(q − 1) = y for infinitely many values of y....

These notes represent an extended version of a talk I gave for the participants of the IMO 2009 and other interested people. We introduce diophantine equations and show evidence that it can be hard to solve them. Then we demonstrate how one can solve a specific equation related to numbers occurring several times in Pascal’s Triangle with state-of-the-art methods. 1. Diophantine Equations The to...

The design of multivariable ripple-free deadbeat controllers is a complex task. One approach which has shown promise for solving multivariable ripple-free deadbeat control (MRFDC) problems is the use of Diophantine equation parameters. The problem of solving robust multivariable ripple free deadbeat with time delays has not been solved. This paper proposes a hybrid two degree of freedom control...

In this paper, we show how to handle linear Diophantine constraints incre-mentally by using several variations of the algorithm by Contejean and Devie (hereafter called ABCD) for solving linear Diophantine systems 4, 5]. The basic algorithm is based on a certain enumeration of the potential solutions of a system, and termination is ensured by an adequate restriction on the search. This algorith...