Bijoy Rahman Arif | Searcharxiv

arXiv2016

An Inductive Proof of Bertrand's Postulate

In this paper, we are going to prove a famous problem concerning prime numbers. Bertrand postulate states that there is always a prime p with n < p < 2n, if n > 1. Bertrand postulate is not a newer one to be proven, in fact, after his assumption and numerical evidence, Chebyshev was the first person who proved it. Subsequently, Ramanujan proved it using properties of Gamma function, and Erdös published a simpler proof with the help of Primorial function, p#. Our approach is unique in the sense that we have used mathematical induction for finding the upper and lower bounds for the second Chebyshev function, and they are even stronger than Ramanujan bounds finding using Gamma function. Otherwise, our approach is similar the way Ramanujan proved it.

math.NT↗

arXiv2011

On the Footsteps to Generalized Tower of Hanoi Strategy

In this paper, our aim is to prove that our recursive algorithm to solve the "Reve's puzzle" (four- peg Tower of Hanoi) is the optimal solution according to minimum number of moves. Here we used Frame's five step algorithm to solve the "Reve's puzzle", and proved its optimality analyzing all possible strategies to solve the problem. Minimum number of moves is important because no one ever proved that the "presumed optimal" solution, the Frame-Stewart algorithm, always gives the minimum number of moves. The basis of our proof is Bifurcation Theorem. In fact, we can solve generalized "Tower of Hanoi" puzzle for any pegs (three or more pegs) using Bifurcation Theorem. But our scope is limited to the "Reve's puzzle" in this literature, but lately, we would discuss how we can reach our final destination, the Generalized Tower of Hanoi Strategy. Another important point is that we have used only induction method to prove all the results throughout this literature. Moreover, some simple theorems and lemmas are derived through logical perspective or consequence of induction method. Lastly, we will try to answer about uniqueness of solution of this famous puzzle.

cs.DM↗