RAMANUJAN NUMBERS OF ORDERS 2, 3, 4, 5 & 7 AND GAUSSIAN INTEGERS

Synopsis & Scholarly Abstract

Mathematics is like a banyan tree and Number Theory (also known as Higher Arithmetic) is a branch of pure mathematics devoted primarily to the study of the natural numbers which are the building blocks of the real number system. In fact, Number Theory is the great and rich intellectual heritage of mankind and essentially a man-made world to meet his ideals of intellectual perfection. For Plato, numbers were “the highest degree of knowledge”. The same idea was taken up by Nicholas Cusanus, for whom “numbers are the best means of approaching divine truths”. All these views go back to Pythagoras, for whom “numbers alone allow us to grasp the true nature of the Universe”. German Mathematician Hermann Minkowski once remarked “Integral numbers are the fountain head of all the mathematics”. No doubt that number is the essence of mathematical calculation and one may discover beautiful patte in numbers. Specifically, integers exhibit fascinating properties, they form sequences and patte s. Recognizing number patte is also an important problem solving skill. It is worth to quote the remark “God created the natural numbers and all else is the work of man” by the German mathematician Leopold Kronecker. An important patte of numbers is the Ramanujan numbers (or R-numbers) named after the great Indian mathematician Srinivasa Ramanujan. The integers that can be expressed as the sum of the squares of two distinct integers in two different ways are called the Second order Ramanujan numbers and for simplicity, written as R numbers. In general, 2 if R is a Ramunujan number of order n, then its numerical relation can be represented as n n R = a +bn=cn+dn n where a, b, c, d are non-zero distinct integers known as base integers and n is a non-zero positive integer. Obviously, there are many R-numbers in each order which are special cases of Diophantine equations covering a wide area in Number Theory. Various characterizations of numbers are illustrated in the book entitled “Ramanujan 2 R Numbers (Mathematical Thoughts and Ideas)” by M. Meyyappan, S. chand & company limited, New Delhi, 1996. Recently, M.A. Gopalan and S. Vidhyalakshmi published a book entitled “Second Order Ramanujan Numbers and Applications”, LAP LAMBERT Academic publishing, Germany, 2017. A careful researcher of patte s in numbers may observe that the numerical relation for Ramanujan numbers is expressed in terms of real integers. A natural question is that arises now is: Whether the numerical relation can be expressed in terms of Gaussian integers for Ramanujan number? The main thrust of this book is to search for Ramanujan numbers in real integers with Gaussian base integers. In particular, the process of obtaining Ramanujan numbers of orders 2, 3, 4, 5 and 7 in real integers with their respective numerical relation in Gaussian integers has been illustrated.