By David Bressoud, Stan Wagon

ISBN-10: 0470412151

ISBN-13: 9780470412152

A direction in Computational quantity idea makes use of the pc as a device for motivation and clarification. The booklet is designed for the reader to quick entry a working laptop or computer and start doing own experiments with the styles of the integers. It provides and explains a few of the quickest algorithms for operating with integers. conventional themes are coated, however the textual content additionally explores factoring algorithms, primality checking out, the RSA public-key cryptosystem, and strange purposes akin to cost digit schemes and a computation of the strength that holds a salt crystal jointly. complicated subject matters contain persevered fractions, Pell's equation, and the Gaussian primes.

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31. Show that the 19th term of the digital sum sequence whose 1st term is 220 and the 10th term of the digital sum sequence whose 1st term is 284 both equal 418. 32. Determine the sum of the digits of the ®rst million positive integers. 33. The sequence a1 , a2 , F F F is called a Kaprekar sequence, denoted by K a1 , if a1 is a positive integer and a k‡1 ˆ ak ‡ sd (ak ), for k . 1, where sd (n) denotes the sum of the digits of n. For example, if a1 ˆ 1, we obtain the Kaprekar sequence K 1 ˆ 1, 2, 4, 8, 16, 23, 28, F F F X In 1959, Kaprekar showed that there are three types of Kaprekar sequence: (I) each term is not divisible by 3, (II) each term is divisible by 3 but not by 9, and (III) each term is divisible by 9.

E. Cohn showed that except for unity, the only square Fibonacci number is 144. 15. Algebraically, let jACj ˆ a and jABj ˆ b; then baa ˆ aa(b À a), hence, b2 À ab ˆ a2 . Dividing both sides 2 of the equation by ap setting x ˆ baa, we obtain x 2 ˆ x ‡ and  p1,  whose roots are ô ˆ (1 ‡ 5)a2, the golden ratio, and ó ˆ (1 À 5)a2, its reciprocal. It is thought by many who search for human perfection that the height of a human body of divine proportion divided by the height of its navel is the golden ratio.

For any a, 1ja. If ajb and ajc then for any integers x and y, aj(bx ‡ cy). If ajb and bjc, then ajc. If a . 0, b . 0, ajb and bja, then a ˆ b. If a . 0, b . 0, and ajb, then a < b. The ®rst two properties follow from the fact that a . 1 ˆ a and a . 0 ˆ 0. In order to establish the third property, suppose that a divides b and c. There exist integers r and s such that ar ˆ b and as ˆ c. Hence, bx ‡ cy ˆ arx ‡ asy ˆ a(rx ‡ sy). Since bx ‡ cy is a mulitple of a, a divides bx ‡ cy. Proofs of the other properties are as straightforward and are left as exercises for the reader.

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A Course in Computational Number Theory by David Bressoud, Stan Wagon

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