# An Introduction to the Theory of Numbers, 5th Edition by Ivan Morton Niven, Herbert S. Zuckerman, Hugh L. Montgomery

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 comprise increased therapy of the binomial theorem, concepts of numerical calculation and a bit on public key cryptography. includes a good set of difficulties.

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Prove that there are infinitely many primes by considering the se22 24 23 quence 2 21 + 1,2 + 1,2 + 1,2 + 1,ยทยทยท. (H) *49. If g is a divisor of each of ab, cd, and ac + bd, prove that it is also a divisor of ac and bd, where a, b, c, d are integers. *50. Show that ( ab, cd) = ~ ~ ~ ~ (a, c)( b, d) ( (a c) , (b d) )( (a c) , (b d) ). *51. Show that 24 is the largest integer divisible by all integers less than its square root. (H) *52. /1/. ) Prove that arbitrary unions of open sets are open, and that finite intersections of open sets are open, so that these open sets define a topology in the usual sense.

Thus if memory is limited (as on a programmable pocket calculator), the former method is preferable, whereas on larger machines it is faster to follow the method above. However, this new method is advantageous only in situations in which both the coefficients of b and of c are desired. 18), only the coefficient of b is needed. 15 or indeed on any previous theorem. 15. Many special cases of the Dirichlet theorem, that is, that there are infinitely many primes in the arithmetic progression a, a + b, a + 2b, ...

We now derive an inequality that is more suggestive of the true state of affairs. 19 For every real number y 1 L p<;y ~ 2, > log log y - 1. p Here it is understood that the sum is over all primes p ~ y. 17. ~ 2, and let ,A/ denote the set of all those positive integers n that are composed entirely of primes p not exceeding y. 3 27 Primes ~ y then n E JY, and thus the sum above includes the sum En <; ylln. Let JY denote the largest integer not exceeding y. By the integral test, If n is a positive integer LN -1 n=l n ~ fN+1 -dx = log (N + l) > log y.