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Solved and Unsolved Problems in Number Theory (CHEL/297)

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The author develops the premise that modern number theory evolved from the ancient Greek preoccupation with two mathematical problems. Searching for the esoteric Perfect Numbers (i.e., whole numbers whose proper divisors sum to the number itself 1+2+3=6) and Diophantine Equations (i.e., finding integral solutions to certain algebraic equations, for example, z^2 = x^2 + y^2). The author calls the later Pythagorianism. The book does a good job of showing how Fermat's Little Theorem, Euler's generalization, and the famous Law of Quadradic Reciprocity developed out of the search for Mersenne Primes, and consequently Perfect Numbers. Again, it is interesting to see how Pythagorianism led to the development of algebraic numbers and eventually to the solution of Fermats "Big" Theorem. Along the way the author elaborates on some of the still unresolved conjectures within number theory. The writing can be a little "dense" at times, so that some parts require a second reading. Overall the book is enjoyable to read and you will gain some insight that won't be gleaned from more standard texts.

an alternate proof to show that combination coeffiecient
C(p^n,r)is divisible by p; where p a prime n & r +ve
integers.
idea first we write the expression of c(p^n,r)
& then we can conclude from the expression that this is
divisible by p (how ?).