Diophantine problems related to polynomial-exponential equations

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K17557
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Gunma University
• Principal Investigator: Miyazaki Takafumi 群馬大学, 大学院理工学府, 准教授 (20706725)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 指数型不定方程式 / Jesmanowicz予想 / ディオファントスの組 / 連立ペル方程式 / Bakerの手法 / ベキ剰余理論

Outline of Final Research Achievements

First, I studied a ternary Diophantine equation expressing that a sum of two powers is equal to a power. In particular, I considered not only the case where each of the base numbers on the equation is fixed, but also the case where one of the three terms is a square and the base numbers of the other terms are fixed. On the former case, I verified the conjecture of Jesmanowicz to be true and had some results on other related unsolved problems. On the latter one, I had some results on Ramanujan-Nagell type equations having very particular conditions on its parameters.
Second, I studied the sets of natural numbers with the property that the product of any two elements in the set increased 1 is a perfect square, so called Diophantine tuples. As joint works of Y.Fujita and M.Cipu, I showed that any given Diophantine triple can only be extended to a Diophantine quadruple (in some sense) at most 8 ways.

Free Research Field

代数学・整数論・不定方程式

Academic Significance and Societal Importance of the Research Achievements

フェルマーの方程式に類似する項数の少ない不定方程式について研究を行った。特に、方程式の各項が累乗数で与えられる場合を扱い、より具体的には、三項型の指数型方程式やRamanujan-Nagell型方程式、さらには二つの特別な線形回帰数列の一致の決定問題に帰着される連立ペル方程式等について考察を行った。これらの研究の多くは、今日の整数論の発展に多大な影響を与えたフェルマーの最終定理の一般化問題「一般型フェルマー予想」に連なる、あるいは深く関わっている。この様な意味で、これらの研究は、整数論のさらなる発展に寄与するものと考えられる。