Discrete Diophantine Equations: Diophantine Equations, Mathematics, Number Theory, Generalizations of Pythagoras Theorem
Misra, Ram Bilas;;;Bajpai, Ranjana
Synopsis "Discrete Diophantine Equations: Diophantine Equations, Mathematics, Number Theory, Generalizations of Pythagoras Theorem"
Diophantine Equations with integer solutions have a long history from ancient Indian and Babylonian Pythagorean triples to the 3rd-century works of Diophantus of Alexandria (DiA) - who studied them systematically for the first time. Ancient Indian scholar Baudhayana (ca. 800 - 500 B.C.) gave a result a millennium before, later called as Pythagoras theorem. He discovered it in the construction of ritual fire altars. Babylonians (c. 1700 B.C.) and Greeks understood Pythagorean triples x2 + y2 = z2, revealing early number theoretic knowledge. DiA, called the ‘father of algebra’, compiled about 200 equations in his Arithmetica seeking rational solutions. Pierre de Fermat of France popularized this study in 17th Century. Later, Euler, Lagrange and Legendre developed methods for solving quadratic forms and Pell’s equation in 18th - 19th Centuries. The present book is a compilation of authors’ recent works exploring integer solutions of various Diophantine Eqs. It consists of 6 chapters of which the first one deals with solutions of a large number of the equations discussed therein. Solutions of exponential Diophantine equations (EDE) involving 4 and 5 terms generalizing Catalan’s conjecture (1844 A.D.) are discussed in Chapters 2 and 3. Various aspects of Brahmagupta’s theorem including Brahmagupta-Fibonacchi identity, different forms of Pell’s equation and Chakravala method of Bhāskara II are dealt in Chapter 4. Applications of Fermat’s little theorem are discussed in Chapter 5. It also deals with miscellaneous puzzles, identities, distribution of 81 cows into 9 groups, and a famous conjecture inserting 4 circles in a square or 8 spherical balls in a cuboid announced in the Mathematical Intelligencer. The solutions of simultaneous EDEs: x = yn, xn = y, x ≠ y for a non-zero integer n form its integral part. The last chapter discusses factorization of numbers of the form 4k2 + 1 and 8k2 + 1, when k is an integer. The book ends in a Bibliography and alphabetical index.
