Small Modular Solutions to Fermat’s Last Theorem  

作　　者：Thomas Beatty Thomas Beatty(Department of Mathematics, Florida Gulf Coast University, Fort Myers, FL, USA)

机构地区：[1]Department of Mathematics, Florida Gulf Coast University, Fort Myers, FL, USA

出　　处：《Advances in Pure Mathematics》2024年第10期797-805,共9页理论数学进展（英文）

摘　　要：The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.

关 键 词：Fermat’s Last Theorem Modular Arithmetic CONGRUENCES Prime Numbers Primitive Roots Indices Ramsey Theory Schur’s Lemma in Ramsey Theory 

分 类 号：O15[理学—数学]