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

The Ks-band Tully-Fisher Relation - A Determination of the Hubble Parameter from 218 ScI Galaxies and 16 Galaxy Clusters

The value of the Hubble Parameter (H0) is determined using the morphologically type dependent Ks-band Tully-Fisher Relation (K-TFR). The slope and zero point are determined using 36 calibrator galaxies with ScI morphology. Calibration distances are adopted from direct Cepheid distances, and group or companion distances derived with the Surface Brightness Fluctuation Method or Type Ia Supernova. Distances are determined to 16 galaxy clusters and 218 ScI galaxies with minimum distances of 40.0 Mpc. From the 16 galaxy clusters a weighted mean Hubble Parameter of H0=84.2 +/-6 km s-1 Mpc-1 is found. From the 218 ScI galaxies a Hubble Parameter of H0=83.4 +/-8 km s-1 Mpc-1 is found. When the zero point of the K-TFR is corrected to account for recent results that find a Large Magellanic Cloud distance modulus of 18.39 +/-0.05 a Hubble Parameter of 88.0 +/-6 km s-1 Mpc-1 is found. A comparison with the results of the Hubble Key Project (Freedman et al 2001) is made and discrepancies between the K-TFR distances and the HKP I-TFR distances are discussed. Implications for Lamda-CDM cosmology are considered with H0=84 km s-1 Mpc-1. (Abridged)Comment: 37 pages including 12 tables and 7 figures. Final version accepted for publication in the Journal of Astrophysics & Astronom