P. G. Walsh

A polynomial-time complexity bound for the computation of the singular part of a Puiseux expansion of an algebraic function

In this paper we present a refined version of the Newton.polygon process to compute the Puiseux expansions of an algebraic function defined over the rational function field. We determine an upper bound for the bit-complexity of computing the singular part of a Puiseux expansion by this algorithm, and use a recent quantitative version of Eisensteins theorem on power series expansions of algebraic functions to show that this computational complexity is polynomial in the degrees and the logarithm of the height of the polynomial defining the algebraic function.

A computational approach for solving y 2 = 1 k + 2 k + ... + x k

We present a computational approach for finding all integral solutions of the equation y2 = 1k + 2k +...+ xk for even values of k. By reducing this problem to that of finding integral solutions of a certain class of quartic equations closely related to the Pell equations, we are able to apply the powerful computational machinery related to quadratic number fields. Using our approach, we determine all integral solutions for 2 ≤ k ≤ 70 assuming the Generalized Riemann Hypothesis, and for 2 ≤ k ≤ 58 unconditionally.

On two classes of simultaneous pell equations with no solutions

In this paper we describe two classes of simultaneous Pell equations of the form x 2 - dy 2 = z 2 - ey 2 = 1 with no solutions in positive integers x,y,z. The proof is elementary and covers the case (d,e) = (8, 5), which was solved by E. Brown using very deep methods.

On a Question of Kaplansky

1. INTRODUCTION. According to [1] and [4], Kaplansky asked for a proof of the following: if a prime p has the representation p = a

A Note on Jacobi Symbols and Continued Fractions

1. INTRODUCTION. It is well known that the continued fraction expansion of a real quadratic irrational is periodic. Here we relate the expansion for √ rs, under the assumption that rX

On powerful numbers

A powerful number is a positive integer n satisfying the property that p2 divides n whenever the prime p divides n; i.e., in the canonical prime decomposition of n, no prime appears with exponent 1. In [1], S.W. Golomb introduced and studied such numbers. In particular, he asked whether (25,27) is the only pair of consecutive odd powerful numbers. This question was settled in [2] by W.A. Sentance who gave necessary and sufficient conditions for the existence of such pairs. The first result of this paper is to provide a generalization of Sentances result by giving necessary and sufficient conditions for the existence of pairs of powerful numbers spaced evenly apart. This result leads us naturally to consider integers which are representable as a proper difference of two powerful numbers, i.e. n=p1−p2 where p1 and p2 are powerful numbers with g.c.d. (p1,p2)=1. Golomb (op.cit.) conjectured that 6 is not a proper difference of two powerful numbers, and that there are infinitely many numbers which cannot be represented as a proper difference of two powerful numbers. The antithesis of this conjecture was proved by W.L. McDaniel [3] who verified that every non-zero integer is in fact a proper difference of two powerful numbers in infinitely many ways. McDaniels proof is essentially an existence proof. The second result of this paper is a simpler proof of McDaniels result as well as an effective algorithm (in the proof) for explicitly determining infinitely many such representations. However, in both our proof and McDaniels proof one of the powerful numbers is almost always a perfect square (namely one is always a perfect square when n≢2(mod4)). We provide in §2 a proof that all even integers are representable in infinitely many ways as a proper nonsquare difference; i.e., proper difference of two powerful numbers neither of which is a perfect square. This, in conjunction with the odd case in [4], shows that every integer is representable in infinitely many ways as a proper nonsquare difference. Moreover, in §2 we present some miscellaneous results and conclude with a discussion of some open questions.

Maximal ranks and integer points on a family of elliptic curves II

We extend a result of Spearman which provides a sufficient condition for elliptic curves of the form y2 = x3 − px, with p a prime, to have Mordell-Weil rank 2. As in Spearman’s work, the condition given here involves the existence of integer points on these curves.

ON A DIOPHANTINE EQUATION OF CASSELS

J.W.S. Cassels gave a solution to the problem of determining all instances of the sum of three consecutive cubes being a square. This amounts to finding all integer solutions to the Diophantine equation

Irreducibility testing over local fields

y^2=3x(x^2+2)

A quantitative version of Runge’s theorem on diophantine equations

. We describe an alternative approach to solving not only this equation, but any equation of the type