论费马大定理中整数/复数/四元数三元组由多项式方程 aⁿ+bⁿ=cⁿ 本身给出的简单整除限制
On the Simple Divisibility Restrictions by Polynomial Equation a n+bn=cn Itself in Fermat Last Theorem for Integer/Complex/Quaternion Triples
AI总结：
本文从方程本身的整除与互素条件出发，聚焦变量 c 而非指数 n，解释 aⁿ+bⁿ=cⁿ 如何筛除解，并评述复数、四元数等情形。
摘要：
本文分析了费马大定理（FLT，1637）中著名方程 aⁿ+bⁿ=cⁿ 的整除限制如何在 n 大于 2 时筛除许多三元组，使其不能成为费马三元组（即解），从而减少费马三元组的基数。在我们的分析中，对正整数（PI）解 (a,b,c,n) 的限制（直至不再有解）并非沿文献中通过正整数指数 n 的限制、以递减集合 {PI}⊃{奇数}⊃{素数}⊃{正则素数} 等方式进行，而是关于在递增集合 {PI} 中的 {pᵏ} 里的 {素数 p} 排除越来越多的正整数 c。文中分析了费马方程中的整除性与互素性质同排除解之间的关系，并展示了 gcd(a,b,c)、gcd(a+b,cⁿ)、gcd(c−a,bⁿ) 和 gcd(c−b,aⁿ) 的同时取值对解基数减少的影响。此外，与历史上依据指数 n 的值来划分 FLT 的文献相反，我们的推导主要关注变量 c 而非变量 n。其中最著名的是已知约有2500年历史、现存的毕达哥拉斯三元组 (a,b,c,n=2)，以及不到400年前作为首批里程碑的已证情形（Gauss 及后来 Euler（1753）证明 n=3 不存在解，Fermat 证明 n=4 不存在解）。众所周知，Wiles 于1995年以抽象迂回的方式证明了 FLT。文中还评述了 n<0、n:=1/m，以及聚焦毕达哥拉斯三元组的复数和四元数 (a,b,c) 情形。四元数上奇次幂的 FLT 不成立。
英文摘要：
The divisibility restrictions in the famous equation a n+bn=cn in Fermat Last Theorem (FLT, 1637) is analyzed how it selects out many triples to be Fermat triple (i.e. solutions) if n greater than 2, decreasing the cardinality of Fermat triples. In our analysis, the restriction on positive integer (PI) solutions ((a,b,c,n) up to the point when there is no more) is not along with restriction on power n in PI as decreasing sets {PI } containing {odd} containing {primes} containing {regular primes}, etc. as in the literature, but with respect to exclusion of more and more c in PI as increasing sets {primes p} in {p k} in {PI}. The divisibility and co-prime property in Fermat equation is analyzed in relation to exclusion of solutions, and the effect of simultaneous values of gcd(a,b,c), gcd(a+b,cn), gcd(c-a,bn) and gcd(c-b,an) on the decrease of cardinality of solutions is exhibited. Again, our derivation focuses mainly on the variable c rather than on variable n, oppositely to the literature in which the FLT is historically separated via the values of power n. Among the most famous are the known, about 2500 years old, existing Pythagorean triples (a,b,c,n=2) and the first milestones as the proved cases (of non-existence as n=3 by Gauss and later by Euler (1753) and n=4 by Fermat) less than 400 years ago. As it is known, Wiles has proved the FLT in 1995 in an abstract roundabout way. The n<0, n:=1/m, as well as complex and quaternion (a,b,c) cases focusing on Pythagoreans are commented. Odd powers FLT over quaternions breaks.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
