AN OVERVIEW ON INTEGERS OF THE FORM \mathbf{6}^\mathbit{n}\ +\ \mathbf{1}

Volume 8, Issue 3, June 2023     |     PP. 97-106      |     PDF (1134 K)    |     Pub. Date: June 11, 2023
DOI: 10.54647/mathematics110403    91 Downloads     151954 Views  

Author(s)

RAJIV KUMAR, Department of Mathematics, D. J. College, Baraut (Baghpat)
SATISH KUMAR, Department of Mathematics, D. N. (PG) College, Meerut
MUKESH KUMAR, Department of Mathematics, Ch, Charan Singh University, Meerut
DUSHIYANT KUMAR, Department of Applied Sciences, Galgotia College of Engg. & Tech., G. Noida, UP, India

Abstract
We pose various congruences on the integers of form 6^n\ +\ 1, n\ \in\ Z_+, which may encourage younger number theorists to do research in number theory and settle new dimensions in this field. We saw that there are only three prime numbers, namely 7,\ 37, and 1297 of form 6^n\ +\ 1, whenever n\ \in\ Z_+-{{2}^km,\ k\geq6,\ m\equiv1(mod\ 2)}, and no one Fermat numbers represent in this form. Moreover, these integers end with seven, like Fermat numbers F_n,\ n\ \geq\ 2. Also, we discussed some congruences with number theoretic functions \sigma,\varphi, and Möbious function \mu, and generates various families of integers with \mu(n)=0.

Keywords
Congruences, Fermat Number, Number Theoretic Functions, Prime Number

Cite this paper
RAJIV KUMAR, SATISH KUMAR, MUKESH KUMAR, DUSHIYANT KUMAR, AN OVERVIEW ON INTEGERS OF THE FORM \mathbf{6}^\mathbit{n}\ +\ \mathbf{1} , SCIREA Journal of Mathematics. Volume 8, Issue 3, June 2023 | PP. 97-106. 10.54647/mathematics110403

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