APPENDIX A generate, create qr codes none on .net projects 2 of 5 Industrial SOME ASPECTS OF NUMBER THEORY A.1 Prime and Relatively Prime Numbers Divisors Prime Numbers Relatively Prime Numbers A.2 Modular Arithmetic.

A.1 / PRIME AND RELATIVELY PRIME NUMBERS The Devil said to Daniel Webster: Set me a task I can t carry out, and I ll give you anything in the world you ask for. Daniel Webster: Fair enough. Prove that for n greater than 2, the equation an + bn = cn has no non-trivial solution in the integers.

They agreed on a three-day period for the labor, and the Devil disappeared. At the end of three days, the Devil presented himself, haggard, jumpy, biting his lip. Daniel Webster said to him, Well, how did you do at my task Did you prove the theorem Eh No .

. . no, I haven t proved it.

Then I can have whatever I ask for Money The Presidency What Oh, that of course. But listen! If we could just prove the following two lemmas . The Mathematical Magpie, Clifton Fadiman In this appendix, we prov ide some background on two concepts referenced in this book: prime numbers and modular arithmetic.. A.1 PRIME AND RELATIVELY PRIME NUMBERS In this section, unless o therwise noted, we deal only with nonnegative integers. The use of negative integers would introduce no essential differences..

Divisors We say that b Z 0 divides a if a = mb for some m, where a, b, and m are integers. That is, b divides a if there is no remainder on division. The notation b a is commonly used to mean b divides a.

Also, if b a, we say that b is a divisor of a. For example, the positive divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The following relations hold: If a 1, then a = ;1.

If a b and b a, then a = ;b. Any b Z 0 divides 0. If b g and b h, then b (mg + nh) for arbitrary integers m and n.

If b g, then g is of the form g = b g1 for some integer g1. If b h, then h is of the form h = b h1 for some integer h1. So mg + nh = mbg1 + nbh1 = b (mg1 + nh1) and therefore b divides mg + nh.

. To see this last point, note that Prime Numbers An integer p > 1 is a visual .net QR Code prime number if its only divisors are ;1 and ;p. Prime numbers play a critical role in number theory and in the techniques discussed in 3.

. APPENDIX A / SOME ASPECTS OF NUMBER THEORY Any integer a > 1 can VS .NET QR Code JIS X 0510 be factored in a unique way as a = pa1 * pa2 * * pat 1 2 t where p1 6 p2 6 6 pt are prime numbers and where each ai is a positive integer. For example, 91 = 7 13 and 11011 = 7 112 13.

It is useful to cast this another way. If P is the set of all prime numbers, then any positive integer can be written uniquely in the following form: a = q pap. where each ap 0 The right-hand side is th qr barcode for .NET e product over all possible prime numbers p; for any particular value of a, most of the exponents ap will be 0. The value of any given positive integer can be specified by simply listing all the nonzero exponents in the foregoing formulation.

Thus, the integer 12 is represented by {a2 = 2, a3 = 1}, and the integer 18 is represented by {a2 = 1, a3 = 2}. Multiplication of two numbers is equivalent to adding the corresponding exponents: k = mn : kp = mp + np for all p. What does it mean, in ter ms of these prime factors, to say that a b Any integer of the form pk can be divided only by an integer that is of a lesser or equal power of the same prime number, pj with j k. Thus, we can say a b : ap bp for all p.
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