论文标题
费马特的最后定理意味着欧几里得的无限素
Fermat's Last Theorem Implies Euclid's Infinitude of Primes
论文作者
论文摘要
我们表明，Fermat的最后一个定理和Schur的组合定理在$ a+b = c $的单色解决方案上意味着存在无限的素数。特别是，对于诸如$ n = 3 $或$ 4 $的小型指数，这给出了欧几里得定理的新证明，例如在这种情况下，Fermat的Last Theorem都有证明不使用无限量的证据。同样，我们讨论了罗斯定理对算术进程，欣德曼定理和无限的拉姆西理论对欧几里得定理的含义。结果，我们看到欧几里得的定理是许多有趣（看似无关的）数学结果的必要条件。
We show that Fermat's last theorem and a combinatorial theorem of Schur on monochromatic solutions of $a+b=c$ implies that there exist infinitely many primes. In particular, for small exponents such as $n=3$ or $4$ this gives a new proof of Euclid's theorem, as in this case Fermat's last theorem has a proof that does not use the infinitude of primes. Similarly, we discuss implications of Roth's theorem on arithmetic progressions, Hindman's theorem, and infinite Ramsey theory towards Euclid's theorem. As a consequence we see that Euclid's Theorem is a necessary condition for many interesting (seemingly unrelated) results in mathematics.
