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Abstract: We will give upper bounds upon the number of integral solutions to binary quartic Thue equations. We will also study the geometric properties of a specific family of binary quartic Thue equations to establish sharper upper bounds. [Copyright &y& Elsevier]

As an application of the method of Thue-Siegel, we will resolve a conjecture of Walsh to the effect that the Diophantine equation aX4 – bY2 = 1, for fixed positive integers a and b, possesses at most two solutions in positive integers X and Y. Since there are infinitely many pairs ( a, b) for which two such solutions exist, this result is sharp. [ABSTRACT FROM AUTHOR]

Abstract: In this article, we study the cyclotomic polynomials of degree with coefficients restricted to the set . By a cyclotomic polynomial we mean any monic polynomial with integer coefficients and all roots of modulus 1. By a careful analysis of the effect of Graeffe''s root squaring algorithm on cyclotomic polynomials, P. Borwein and K.K. Choi gave a complete characterization of all cyclotomic polynomials with odd coefficients. They also proved that a polynomial with coefficients ±1 of even degree is cyclotomic if and only if , where and the are primes, not necessarily distinct. Here is the pth cyclotomic polynomial. Based on substantial computation, they also conjectured that this characterization also holds for polynomials of odd degree with ±1 coefficients. We consider the conjecture for odd degree here. Using Ramanujan''s sums, we solve the problem for some special cases. We prove that the conjecture is true for polynomials of degree with odd prime p or separable polynomials of any odd degree. [Copyright &y& Elsevier]