تقرير
A p-adic identity for Wieferich primes
| العنوان: | A p-adic identity for Wieferich primes |
|---|---|
| المؤلفون: | Chua, Kok Seng |
| سنة النشر: | 2021 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Number Theory, 11A05, 11A51 |
| الوصف: | Let $n$ be a positive integer, $p$ be an odd prime and integers $a,b \not= 0$ with $gcd(a,b)=1$, $p \nmid ab$, and $p|(a^n \pm b^n)$, we prove the identity $$\nu_p(a^n \pm b^n)-\nu_p(n)=\nu_p(a^{p-1}-b^{p-1}).$$ An unintended interesting immediate consequence is the following variant of Wieferich's criterion for FLT : Let $x^n+y^n=z^n$ with $n$ prime and $x,y,z$ pairwise relatively prime. Then every odd prime $p|y$ satisfies $\nu_p(z^{p-1}-x^{p-1}) \ge n-1$ and every odd prime $p|x$ satisfies $\nu_p(z^{p-1}-y^{p-1}) \ge n-1$, and every odd prime $p|z$ satisfies $\nu_p(x^{p-1}-y^{p-1}) \ge n-1$, ie. every odd prime dividing $xyz$ is a Wieferich prime of order at least $n-1$ to some base pair. In the "first case" where $n \nmid xyz$, the lower bound for the Wieferich order can be improved to $n$. This gives us very strong intuition why there should not be any solution even for moderately large $n$. Comment: 5 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2112.04173 |
| رقم الأكسشن: | edsarx.2112.04173 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |