Monday, June 24, 2019

Number Theory as an Adventure

Number theory can be taught to children. It all begins with integers. First year algebra (at least) is required to prove these conjectures. But the facts can be presented and tested by a nine-year old who is comfortable with long division. Even a first grader can understand, appreciate, and test many of them. 

Multiplying odds and evens always yields: odd X odd = odd; even X even = even; odd X even = even. A child learning arithmetic can test examples until she is tired and at least get a lot of practice. It is not a proof, of course, but it can remain a challenge for a child who wants to learn algebra. The algebra is easy to show. 
An Adventurer's Guide to Number Theory
by Richard Friedberg, Dover, 1968, 1994
(This Dover edition contains corrections
and appendices not in the McGraw-Hill
printing of 1968.)
A child who is comfortable with first year algebra can get lost in the Pythagorean theorem and many of the truths that can be derived from it. Among the Pythagorean Triples are some such as 3-4-5 and 5-12-13 and 15-112-113 for which the hypotenuse is one unit longer than the longer of the two other sides. And there is an easy formula for generating them. It can be coded up as a computer program or a spreadsheet. 

This second edition was released soon after Andrew Wiles announced his proof of Fermat’s Last Theorem. Richard Friedberg cautions that the proof was not yet accepted. It was not. A second proof was. Friedberg discusses Fermat’s Last Theorem as an extension of other explorations, principally the Pythagorean Theorem. That venerable truth is the crux of this work though Friedberg brings the reader into many other fields: infinite descent, perfect numbers, modulo arithmetic, and prime numbers. Along the way, he cites the discoveries made by Fermat, Euler, Lagrange, Gauss, and, of course, Diophantus, among others. 


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