Ijraset Journal For Research in Applied Science and Engineering Technology

Abstract

In this document, the range of appearance I(r) of a positive integer r in heptagonal numbers and some results by comparing I(r) with Legendre symbol, p- adic range and prime conjunction function of r are accessible. Since I(r) does not possess a meticulous prototype, all the results are confirmed by Java program for all-natural numbers r?N.

Introduction

Conclusion

In this paper, the range of appearance I(r) of a positive integer r in heptagonal numbers is defined and limited number of results consisting I(r), legendre symbol, p- adic range and prime conjunction function of r are evaluated. Entire results are inveterate by Java program for all positive values of r. In this approach, one can define the range of appearance of an integer in other polygonal numbers and may analyse their results by comparing the values of I(r) with Jacobi symbol, quadratic residue and non-residue modulo r.

References

Ivan Niven, Hibert S, Zuckerman and Huge L, Montgomery, “An Introduction to the Theory of Numbers”, Fifth Edn., John Wiley & Sons Inc., 2004. Carmichael, R.D., “The Theory of Numbers and Diophantine Analysis”, Dover Publication, New York (1959). Mordell, L.J., “Diophantine Equations”, Academic press, London (1969). John,H., Conway and Richard K.Guy, “The Book of Numbers”, Springer Verlag, NewYork,1995. Eva Trojovs, “On the Diophantine Equation z(n)=(2-1/k)n involving the order of appearance in the Fibonacci Sequence”, Mathematics 2020,1-8. Trojovský, P. “The order of appearance of the sum and difference between two Fibonacci Numbers”,Asian-Eur.J.Math.2019,12,1950046. Chung, C.L. “Some Polynomial Sequence Relations”. Mathematics 2019, 7,750. Knott, R. “Fibonacci Numbers and the Golden Section”, The Mathematics Department of the University of Surrey, UK. Available online:Http://www.maths,surrey.ac.uk/hostedsites/R.Knott/Fibonacci/fib.html . (accessed on 2 January 2020). The Fibonacci Association, Official Website. Available online: https://www.mathstat.dal.ca/fibonacci/(accessed on 2 January 2020). Kim, S. “The density of the terms in an elliptic divisibility sequence having a fixed G.C.D. with their indices”. J. Number Theory2020,207, 22–41.

Copyright

Copyright © 2022 S Saranya, V. Pandichelvi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.