Some Unpublished Works of (Late) Dr. S.P. Khare

Abstract

The senior author had a long association with Dr. Satgur Prasad Khare, as his student both in graduate and postgraduate classes at the University of Allahabad, Prayagraj (India) for over five decades. Dr. Khare worked in the Theory of Numbers under the supervision of a celebrated Indian Number Theorist (Prof. H. Gupta). Even after his retirement, he continued his researches in Number Theory and, recently, he developed interest in cryptography. Due to lack of senior workers in his field, he trusted his own teacher and often used to send his researches to the author for review. He met a tragic and premature end of his life on 30 April, 2021 leaving a lot of his unpublished work that needs to be highlighted. He was completing a research monograph on Number Theory before his sudden cardiac arrest. The present article is a compilation of some of his bizarre researches, which he had been sending to the author. An attempt is made to edit and revise the works of Dr. Khare and is presented here. It consists of certain recurrence relations for square and cubes of large integers and other interesting problems detailed below. Further works of Dr. Khare dwelling upon the “Primality and factorization” of RSA numbers will be presented in a series of papers.