应用费尔马小定理求解一些同余和不定方程

Using Fermat’s Little Theorem to Solve Some Congruence Equations and Indefinite Equations

摘要: 本文主要利用费尔马小定理研究一些同余方程和不定方程解的问题。根据费尔马小定理证明问题的条件和思想，通过详细的推导得到了一些重要的结论。这些结论主要包括21x¹⁸+2y¹⁵-x⁴-3≡0(mod7)和x²+3≡0(mod5)无整数解；不定方程x³-3xy²+y³=2981和15x²-7y²=9无整数解。此外，还利用费尔马小定理考虑了一些多项式问题和五次不定方程x⁵+y⁵=z⁵的解，并给出了其他的一些应用。

Abstract: In this paper, using Fermat’s little theorem, the solutions of some congruence equations and in-definite equations are mainly studied. Through the idea of Fermat’s small theorem in the proof, some important conclusions are obtained according to the detailed derivation. These conclusions are mainly given. The equations 21x¹⁸+2y¹⁵-x⁴-3≡0(mod7) and x²+3≡0(mod5) have not integer solutions. The indefinite equations x³-3xy²+y³=2981 and 15x²-7y²=9 have not inte-ger solutions. In addition, some polynomial problems are also considered. And the solutions of pentadic indefinite equation x⁵+y⁵=z⁵ are investigated by using Fermat’s small theorem. Finally, the others applications of the Fermat’s small theorem are given.