ARTICLE | doi:10.20944/preprints202108.0176.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: consecutive sum of the digits; algebraic equations; diophantine equations; arithmetic functions
Online: 9 August 2021 (07:52:49 CEST)
In this paper defines the consecutive sum of the digits of a natural number, so far as it becomes less than ten, as an arithmetic function called and then introduces some important properties of this function by proving a few theorems in a way that they can be used as a powerful tool in many cases. As an instance, by introducing a test called test, it has been shown that we are able to examine many algebraic equalities in the form of in which and are arithmetic functions and to easily study many of the algebraic and diophantine equations in the domain of whole numbers. The importance of test for algebraic equalities can be considered equivalent to dimensional equation in physics relations and formulas. Additionally, this arithmetic function can also be useful in factorizing the composite odd numbers.
ARTICLE | doi:10.20944/preprints202009.0742.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Prime numbers; lemmas and theorems; Fermat’s factorization method; sieve
Online: 30 September 2020 (12:12:58 CEST)
For each non-prime odd number as F=pq , if we consider m/n as an approximation for q/p and choose k=mn , then by proving some lemmas and theorems, we can compute the values of m and n. Finally, by using Fermat’s factorization method for F and 4kF as difference of two non-consecutive natural numbers, we should be able to find the values of p and q. Then we introduce two new and powerful sieves for separating composite numbers from prime numbers.