Fermat's Lemma on the Hyperbolic Plane

Generalizing classical mathematical problems to other spaces is an important research direction in mathematics. This paper studies Fermat's Lemma on the hyperbolic plane, where the set of hyperbolic numbers is a commutative ring with zero divisors generated by two real numbers, similar to the complex numbers but more complicated. Compared to the classical Fermat’s Lemma, the Fermat’s Lemma in this paper has a higher dimension and is built on a hyperbolic ring whose algebraic structure is more complex than the real number field. On the basis of a generalization of the classical Fermat’s Lemma, the paper overcomes the difficulty that hyperbolic numbers contain zero divisors, and obtains the proof of the first and second Fermat’s Lemma on the hyperbolic plane through the decomposition of hyperbolic numbers, which will give impetus to theoretical research in hyperbolic analysis.