论文标题
反映各种类型的数字，我
Reflecting Numbers of Various Types, I
论文作者
论文摘要
本文的目的是将反映数字的概念介绍给数字理论领域，并对反映某些类型的数量进行分类。对于我们来说，反映数字来自一致的数字，以上一致的数字，并且远离一致的数字。 明确地说，反映$ $（k，m）$的类型数量是两个不同的有理$ k $ th的平均值，距离之间的距离是另一个非零有理$ m $ th的两倍。特别是，反映$（2,2）$的类型数量都是一致的数字，因此将被称为反映本文中的一致数字。我们可以证明，所有质数$ p \ equiv5 \ mod8 $都反映了一致，总的来说，对于任何整数$ k \ ge0 $，都有无限的无平方英尺反映$ 5 $ modulo $ 8 $的一致性，$ 5 $ modulo $ 8 $，与$ k+1 $ 1 $ prime divisors。此外，我们猜想所有素数的数字$ p \ equiv1 \ mod8 $都反映了一致。此外，我们表明，如果$ \ gcd（k，m）\ ge3 $，则没有反映$ $（k，m）$的数量。
The purpose of this paper is to introduce the concept of reflecting numbers to the realm of number theory and to classify reflecting numbers of certain types. For us, reflecting numbers are coming from congruent numbers, above congruent numbers, and away from congruent numbers. Explicitly speaking, a reflecting number of type $(k,m)$ is the average of two distinct rational $k$th powers, between which the distance is twice another nonzero rational $m$th power. In particular, reflecting numbers of type $(2,2)$ are all congruent numbers and thus will be called reflecting congruent numbers in this paper. We can show that all prime numbers $p\equiv5\mod8$ are reflecting congruent and in general for any integer $k\ge0$ there are infinitely many square-free reflecting congruent numbers in the residue class of $5$ modulo $8$ with exactly $k+1$ prime divisors. Moreover, we conjecture that all prime congruent numbers $p\equiv1\mod8$ are reflecting congruent. In addition, we show that there are no reflecting numbers of type $(k,m)$ if $\gcd(k,m)\ge3$.
