|
|
数论函数方程的可解性
|
Abstract:
本文主要研究数论函数方程Z(n)=φ7(SL(n))的可解性。为此，先给出广义欧拉函数φ7(n)的表达式。由此，给出φ7(pβ)的表达式，其中p是素数，且β∈？+。最后，讨论该方程的可解性，我们证明了其无正整数解。
In this paper, we mainly study the solvability of the arithmetic equationZ(n)=φ7(SL(n)). For this purpose, we derive the expression of the generalized Euler functionφ7(n), from which the formula ofφ7(pβ)is obtained, where p is prime, andβ∈？+. Afterwards, the solvability of the above equation is discussed, thus drawing the conclusion that it is has no solution in positive integers.
| [1] | Lehmer, E. (1938) On Congruences Involving Bernoulli Numbers and the Quotients of Fermat and Wilson. The Annals of Mathematics, 39, 350-360. https://doi.org/10.2307/1968791 |
| [2] | Ribenboim, P. (1979) 13 Lectures on Fermat’s Last Theorem. Springer-Verlag. |
| [3] | Cai, T. (2002) A Congruence Involving the Quotients of Euler and Its Applications (I). Acta Arithmetica, 103, 313-320. |
| [4] | Cai, T.X., Fu, X.D. and Zhou, X. (2007) A Congruence Involving the Quotients of Euler and Its Applications (II). Acta Arithmetica, 130, 203-214. |
| [5] | Cai, T.X., Shen, Z.Y. and Hu, M.J. (2013) On the Parity of the Generalized Euler Function. Advances in Mathematics (China), 42, 505-510. |
| [6] | Shen, Z.Y., Cai, T.X. and Hu, M.J. (2016) On the Parity of the Generalized Euler function (II). Advances in Mathematics (China), 45, 509-519. |
| [7] | Zhu, C. and Liao, Q.Y. (2022) A Recursion Formula for the Generalized Euler Function . https://doi.org/10.48550/arXiv.2105.10870 |
| [8] | 贺艳峰, 李勰, 韩帆, 薛媛媛. 数论函数方程的可解性研究[J]. 湖北大学学报(自然科学版), 2023: 1-8. http://kns.cnki.net/kcms/detail/42.1212.N.20231018.1016.004.html, 2024-03-31. |
| [9] | 王慧莉, 廖群英, 任磊. 方程的可解性[J]. 四川师范大学学报(自然科学版), 2022, 45(5): 595-604. |
| [10] | 姜莲霞, 张四保. 有关广义欧拉函数的一方程的解[J]. 首都师范大学学报(自然科学版), 2020, 41(6): 1-5. |
| [11] | 张四保. 数论函数方程的可解性[J]. 西南大学学报(自然科学版), 2020, 42(4): 65-69. |
| [12] | 朱杰, 廖群英. 方程的可解性[J]. 数学进展, 2019, 48(5): 541-554. |
