Elementary Number Theory Problems. Part II

IF 1 Q1 MATHEMATICS

Formalized Mathematics Pub Date : 2021-04-01 DOI:10.2478/forma-2021-0006

Artur Korniłowicz, Dariusz Surowik

引用次数: 1

Abstract

Summary In this paper problems 14, 15, 29, 30, 34, 78, 83, 97, and 116 from [6] are formalized, using the Mizar formalism [1], [2], [3]. Some properties related to the divisibility of prime numbers were proved. It has been shown that the equation of the form p2 + 1 = q2 + r2, where p, q, r are prime numbers, has at least four solutions and it has been proved that at least five primes can be represented as the sum of two fourth powers of integers. We also proved that for at least one positive integer, the sum of the fourth powers of this number and its successor is a composite number. And finally, it has been shown that there are infinitely many odd numbers k greater than zero such that all numbers of the form 22n + k (n = 1, 2, . . . ) are composite.

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初等数论问题。第二部分

本文利用Mizar形式主义[1]，[2]，[3]，对[6]中的问题14、15、29、30、34、78、83、97和116进行形式化。证明了素数可除性的一些性质。证明了p2 + 1 = q2 + r2式(其中p, q, r为质数)至少有四个解，并证明了至少有五个质数可以表示为两个整数的四次幂的和。我们还证明了对于至少一个正整数，这个数及其后继数的四次方之和是合数。最后，证明了有无穷多个大于零的奇数k，使得所有形式为22n + k (n = 1,2，…)的数是合成的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Formalized Mathematics MATHEMATICS-

自引率

0.00%

发文量

审稿时长

10 weeks

期刊介绍： Formalized Mathematics is to be issued quarterly and publishes papers which are abstracts of Mizar articles contributed to the Mizar Mathematical Library (MML) - the basis of a knowledge management system for mathematics.