<abstract xmlns="http://www.w3.org/1999/xhtml">

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 <italic>p</italic>2 + 1 = <italic>q</italic>2 + <italic>r</italic>2, where <italic>p</italic>, <italic>q</italic>, <italic>r</italic> 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 <italic>k</italic> greater than zero such that all numbers of the form 22<italic>n</italic> + <italic>k</italic> (<italic>n</italic> = 1, 2, . . . ) are composite.
</abstract>

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.

Elementary Number Theory Problems. Part II

Formalized Mathematics

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

{"article-title":"Elementary Number Theory Problems. Part II"}

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...

Elementary Number Theory Problems. Part II

Published Online: Aug 26, 2021

Page range: 63 - 68

Accepted: Mar 30, 2021

DOI: https://doi.org/10.2478/forma-2021-0006

Keywords
number theory, divisibility, primes

© 2021 Artur Korniłowicz et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

Elementary Number Theory Problems. Part II

Published Online: Aug 26, 2021

Page range: 63 - 68

Accepted: Mar 30, 2021

DOI: https://doi.org/10.2478/forma-2021-0006

Keywordsnumber theory, divisibility, primes

© 2021 Artur Korniłowicz et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

Keywords
number theory, divisibility, primes