On Polygonal Square Triangular Numbers II

A pentagonal square triangular number is a number which is a pentagonal number, a square and a triangular number at the same time. In our previous paper [10], we have shown the only pentagonal square triangular number is 1. In this note, we shall continue to investigate several related problems and give more detailed results on these subjects

Length of Integer Solutions

We shall introduce a length of the integer solutions of linear diophantine equations and investigate the fundamental properties of this length. We will also give an application of this length to a famous mathematical puzzle called three jug problem

Corner the Knight Game

In this paper, we shall introduce a game called Corner the Knight Game, which is a game changed the chess queen of Corner the Queen Game to the chess knight. Although this game is very simple, we could not find any literature on this game. Hence it should be of some meaning to investigate this elementary game

Square, Rectangular and Triangular Nim Games

Let p be an integer with p ≥ 2. We shall investigate the following two piles Nim games. Let S be the set of positive integers {1 ≤ i ≤ p − 1}. Each player can remove the number of tokens s1 ∈ {0} ∪ S from the first pile and s2 ∈ {0} ∪ S from the second pile with 0 < s1 + s2 at the same time. We shall identify (m, n) to a position of this Nim game, where m is the number of tokens in the first pile and n is the number of tokens in the second pile. We shall show the Sprague-Grundy sequence (or simply G-sequences) gs(m, n) satisfy the periodic relation gs(m+p, n+p) = gs(m, n) for any position (m, n). We will call this two piles Nim Square Nim. In case m and n are sufficiently large, we will show that G-sequences gs(m, n) are also periodic for each row and column with the same period p. Finally we shall introduce several related games, such as Rectangular Nim, Triangular Nim and Polytope Nim

Modified Farey Trees and Pythagorean Triples

In 1963, F. J. M. Barning discovered a ternary tree of primitive Pythagorean triples, where each triple is transformed to other three triples by three distinct 3 × 3 unimodular matrices. This fact has been rediscovered many times. In this paper, we shall give an elementary explanation of this fact using classical Euclidean parametrization of primitive Pythagorean triples and modified ternary Farey trees

On the Class Numbers of Real Quadratic Fields of Richaud-Degert Type

The purpose of this paper is to give an explicit proof of the infinity of real quadratic fields of Richaud-Degert type and construct the infinite sequences of real quadratic fields with large class numbers

On Some Formulas for π/2

In his paper [1], J.G. Goggins has shown a formula which relates π and Fibonacci numbers. In our paper [2]，we have proved a generalized version of this formula. In this note, we shall prove formulas which generalize Fibonacci number to certain binary recurrence sequences

Several Methods for Solving Simultaneous Fermat-Pell Equations

In our previous papers [12] and [13] , we have exhibited the structure of certain real bicyclic biquadratic fields and as a byproduct solved the simultaneous Fermat-Pell equations x^2-3y^2=1, y^2-2z^2=-1 have only one non-negative integer solution: (x, y, z)=(2, 1, 1). In this paper, we shall investigate similar simultaneous Fermat-Pell equations and solve them by several different methods

Hypothesis of Schinzel and Sierpiński and Cyclotomic Fields with Isomorphic Galois Groups

In 1922 R. D. Carmichael conjectured that for any natural number n there exist infinitely many natural numbers m such that φ(n) = φ(m). It is well known that this conjecture can be proved under the assumption of the famous unproved hypothesis of Schinzel and Sierpiński. In this short note, we shall show the Hypothesis of Schinzel and Sierpiński implies more precisely that the existence of infinitely many cyclotomic fields Q(ζn) and Q(ζm) with isomorphic absolute Galois groups. Here ζn and ζm are primitive nth and mth roots of unity with m ≠ n

Unit Groups of Some Quartic Fields

Using the methods developed in [5] and J.H.E.Cohn's results on some quartic diophantine equation, we shall show the structure of the unit groups of the quartic fields Q(√+4>,√+16>), where F_m and L_m are the mth Fibonacci and Lucas numbers. At the same time, we shall show the explicit class number formulae for these quartic fields