By Ivan Morton Niven, Herbert S. Zuckerman, Hugh L. Montgomery

The 5th version of 1 of the traditional works on quantity thought, written through internationally-recognized mathematicians. Chapters are fairly self-contained for better flexibility. New positive aspects contain elevated therapy of the binomial theorem, suggestions of numerical calculation and a piece on public key cryptography. comprises a great set of difficulties.

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**Additional resources for An Introduction to the Theory of Numbers, 5th Edition **

**Example text**

9. 3 A reduced residue system modulo m is a set of integers r; such that (r;, m) = 1, r; ;;j= rj (mod m) if i j, and such that every x prime to m is congruent modulo m to some member r; of the set. 4 it is clear that a reduced residue system modulo m can be obtained by deleting from a complete residue system modulo m those members that are not relatively prime to m. Furthermore, all reduced residue systems modulo m will contain the same number of members, a number that is denoted by

Bb! I(ab)!. (H) 6. Let f(x) and g(x) be n-times differentiable functions. Show that the nth derivative of f(x)g(x) is n t (~)f(k)( x) g(n-k)( x). k~O 7. Show that ( - \- 1 ) = (_l)k( a ; Izl < 1 then 1 (1 - z)"'+! k) for k~ O. Deduce that if =;. 15) 8. Give three proofs that M L m=O (m+k)=(k+M+l) k +1 k . 20. /= {l,2,"', k + M + l}. Count the number of subsets N of . / containing k + 1 elements, with the maximum one being k+m+1. (c) Compute the coefficient of (1 ZM in the identity 1 1 (1 - Z)k+!

Exhibit three integers that are relatively prime but not relatively prime in pairs. 8. Two integers are said to be of the same parity if they are both even or both odd; if one is even and the other odd, they are said to be of opposite parity, or of different parity. Given any two integers, prove that their sum and their difference are of the same parity. 9. Show that if ae Ibe then a lb. 10. Given a Ib and e Id, prove that ae Ibd. 11. Prove that 4%(n 2 + 2) for any integer n. 12. Given that (a, 4) = 2 and (b, 4) = 2, prove that (a + b,4) = 4.