Diophantine problems related to polynomial-exponential equations
| Project/Area Number |
16K17557
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Algebra
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| Research Institution | Gunma University |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| Budget Amount *help |
¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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| 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 / 単数方程式 / 寺井予想 / 線形回帰数列 / Ramanujan-Nagell 方程式 / 連立ぺル方程式 |
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| 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.
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| 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。
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Report
(5 results)
Research Products
(17 results)
