Squares from products of integers

This is a preprint of an article published in the Gazette of the Australian Mathematical Society, 31 (2004) no.1, pp.40-42.Notice that 1_2_3_4+1 = 52 , 2_3_4_5+1 = 112 , 3_4_5_6+1 = 192 , . . . . Indeed, it is well known that the product of any four consecutive integers always differs by one from a perfect square. However, a little experimentation readily leads one to guess that there is no integer n, other than four, so that the product of any n consecutive integers always differs from a perfect square by some fixed integer c = c(n) depending only on n. The two issues that are present here can be readily dealt with. The apparently special status of the number four arises from the fact that any quadratic polynomial can be completed by a constant to become the square of a polynomial. Second, [5] provides an elegant proof that there is in fact no integer n larger than four with the property stated above. In [5] one finds a reminder that a polynomial taking too many square values must be the square of a polynomial (see [4, Chapter VIII.114 and .190], and [2]). One might therefore ask whether there are polynomials other than integer multiples of x(x + 1)(x + 2)(x + 3) and 4x(x + 1), with integer zeros and differing by a nonzero constant from the square of a polynomial. We will show that this is quite a good question in that it has a nontrivial answer, inter alia giving new insight into the results of [5]

UNAMBIGUOUS EVALUATIONS OF BIDECIC JACOBI AND JACOBSTHAL SUMS

Abstract For a class of primes p = 1 (mod 20) for which 2 is a quintic nonresidue, unambiguous evaluations of parameters of bidecic Jacobi and Jacobsthal sums (modp

An elementary proof of the irrationality of Tschakaloff series

We present a new proof of the irrationality of values of the series $T_q(z)=\sum_{n=0}^\infty z^nq^{-n(n-1)/2}$ in both qualitative and quantitative forms. The proof is based on a hypergeometric construction of rational approximations to $T_q(z)$.Comment: 5 pages, AMSTe

ZnO薄膜を用いた光導波路型波長変換デバイスの設計と作製に関する研究

修士論

On Local Behavior of Holomorphic Functions Along Complex Submanifolds of C^N

In this paper we establish some general results on local behavior of holomorphic functions along complex submanifolds of \Co^{N}. As a corollary, we present multi-dimensional generalizations of an important result of Coman and Poletsky on Bernstein type inequalities on transcendental curves in \Co^{2}.Comment: minor changes in the formulation and the proof of Lemma 8.