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Unformatted text preview: 4 mcs-ftl 2010/9/8 0:40 page 81 #87 Number Theory Number theory is the study of the integers. Why anyone would want to study the integers is not immediately obvious. First of all, whats to know? Theres 0, theres 1, 2, 3, and so on, and, oh yeah, -1, -2, . . . . Which one dont you understand? Sec- ond, what practical value is there in it? The mathematician G. H. Hardy expressed pleasure in its impracticality when he wrote: [Number theorists] may be justified in rejoicing that there is one sci- ence, at any rate, and that their own, whose very remoteness from or- dinary human activities should keep it gentle and clean. Hardy was specially concerned that number theory not be used in warfare; he was a pacifist. You may applaud his sentiments, but he got it wrong: Number Theory underlies modern cryptography, which is what makes secure online communication possible. Secure communication is of course crucial in warwhich may leave poor Hardy spinning in his grave. Its also central to online commerce. Every time you buy a book from Amazon, check your grades on WebSIS, or use a PayPal account, you are relying on number theoretic algorithms. Number theory also provides an excellent environment for us to practice and apply the proof techniques that we developed in Chapters 2 and 3 . Since well be focusing on properties of the integers, well adopt the default convention in this chapter that variables range over the set of integers , Z . 4.1 Divisibility The nature of number theory emerges as soon as we consider the divides relation a divides b iff ak D b for some k: The notation, a j b , is an abbreviation for a divides b . If a j b , then we also say that b is a multiple of a . A consequence of this definition is that every number divides zero. This seems simple enough, but lets play with this definition. The Pythagoreans, an ancient sect of mathematical mystics, said that a number is perfect if it equals the sum of its positive integral divisors, excluding itself. For example, 6 D 1 C 2 C 3 and 28 D 1 C 2 C 4 C 7 C 14 are perfect numbers. On the other hand, 10 is not perfect because 1 C 2 C 5 D 8 , and 12 is not perfect because 1 C 2 C 3 C 4 C 6 D 16 . 82 mcs-ftl 2010/9/8 0:40 page 82 #88 Chapter 4 Number Theory Euclid characterized all the even perfect numbers around 300 BC. But is there an odd perfect number? More than two thousand years later, we still dont know! All numbers up to about 10 300 have been ruled out, but no one has proved that there isnt an odd perfect number waiting just over the horizon. So a half-page into number theory, weve strayed past the outer limits of human knowledge! This is pretty typical; number theory is full of questions that are easy to pose, but incredibly difficult to answer....
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