# Elementary Number Theory in Nine Chapters, Second Edition by James J. Tattersall

By James J. Tattersall

Meant to function a one-semester introductory direction in quantity conception, this moment variation has been revised all through. particularly, the sector of cryptography is highlighted. on the center of the publication are the foremost quantity theoretic accomplishments of Euclid, Fermat, Gauss, Legendre, and Euler. furthermore, a wealth of latest workouts were integrated to completely illustrate the houses of numbers and ideas constructed within the textual content. The e-book will function a stimulating creation for college kids new to quantity conception, despite their historical past. First version Hb (1999) 0-521-58503-1 First version Pb (1999) 0-521-58531-7

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2 without proof. 2 (Alternate principle of mathematical induction) Any set of natural numbers that contains the natural number m, and contains n þ 1 whenever it contains all the natural numbers between m and n, where n > m, contains all the natural numbers greater than m. The alternate principle of mathematical induction implies the well-ordering principle. In order to see this, let S be a nonempty set of natural numbers with no least element. For n . 1, suppose 1, 2, . . , n are elements S, the complement of S.

Lucas originally deﬁned v n to be u2 n =un . He derived many relationships between Fibonacci and Lucas numbers. For example, u nÀ1 þ u nþ1 ¼ v n , un þ v n ¼ 2u nþ1 , and v nÀ1 þ v nÀ1 ¼ 5un . The sequence of Lucas numbers is an example of a Fibonaccitype sequence, that is, a sequence a1 , a2 , . . , with a1 ¼ a, a2 ¼ b, and a nþ2 ¼ a nþ1 þ an , for n > 2. Fibonacci numbers seem to be ubiquitous in nature. There are abundant references to Fibonacci numbers in phyllotaxis, the botanical study of the arrangement or distribution of leaves, branches, and seeds.

The tribonacci numbers an are deﬁned recursively as follows: a1 ¼ a2 ¼ 1, a3 ¼ 2, and an ¼ a nÀ1 þ a nÀ2 þ a nÀ3, for n > 4. Generate the ﬁrst 20 tribonacci numbers. 22. The tetranacci numbers bn are deﬁned as follows: b1 ¼ b2 ¼ 1, b3 ¼ 2, b4 ¼ 4, and bn ¼ b nÀ1 þ b nÀ2 þ b nÀ3 þ b nÀ4, for n > 5. Generate the ﬁrst 20 tetranacci numbers. 23. Verify the Collatz conjecture for the following numbers: (a) 9, (b) 50, (c) 121. 24. Determine the three cycles that occur when 3an À 1 is substituted for 3an þ 1 in the Collatz algorithm.