%0 Journal Article
%T 应用费尔马小定理求解一些同余和不定方程<br>Using Fermat’s Little Theorem to Solve Some Congruence Equations and Indefinite Equations
%A 苑金臣
%A 郭艳凤
%A 王军霞
%J Pure Mathematics
%P 3439-3446
%@ 2160-7605
%D 2023
%I Hans Publishing
%R 10.12677/PM.2023.1312356
%X 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21x<sup>18</sup>+2y<sup>15</sup>-x<sup>4</sup>-3≡0(mod7)和x<sup>2</sup>+3≡0(mod5)无整数解；不定方程x<sup>3</sup>-3xy<sup>2</sup>+y<sup>3</sup>=2981和15x<sup>2</sup>-7y<sup>2</sup>=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x<sup>5</sup>+y<sup>5</sup>=z<sup>5</sup>的解，并给出了其他的一些应用。<br />
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 21x<sup>18</sup>+2y<sup>15</sup>-x<sup>4</sup>-3≡0(mod7) and x<sup>2</sup>+3≡0(mod5) have not integer solutions. The indefinite equations x<sup>3</sup>-3xy<sup>2</sup>+y<sup>3</sup>=2981 and 15x<sup>2</sup>-7y<sup>2</sup>=9 have not inte-ger solutions. In addition, some polynomial problems are also considered. And the solutions of pentadic indefinite equation x<sup>5</sup>+y<sup>5</sup>=z<sup>5</sup> are investigated by using Fermat’s small theorem. Finally, the others applications of the Fermat’s small theorem are given.
%K 费马小定理，同余方程，不定方程，整数解<br>Fermat’s Little Theorem
%K Congruence Equations
%K Indefinite Equations
%K Integer Solutions
%U http://www.hanspub.org/journal/PaperInformation.aspx?PaperID=77651