15,16b 14. It has been conjectured that every even integer can be written as the difference of tb > 1, Theorem 3.2 ensures that b has at least one prime factor p. Then p 1 is composite, then it may be written as a = bc, where 1 1, it therefore suffices to divide1 1 is a square if and only if in the canonical form of aall the exponents of the primes are even integers.16. An integer is said to be square-free if it is not divisible by the square of any integer greaterthan 1. Prove the following:(a) An integer n> 1 is square-free if and only if n can be factored into a product ofdistinct primes.(b) Every integer n > 1 is the product of a square-free integer and a perfect square.[Hint: If n =29; +ri where r¡ = 0 or 1 according as k; is even or odd.]ki „k2PP p is the canonical factorization of n, then write k; =17. Verify that any integer n can be expressed as n = 2km, where k >0 and m is an oddinteger.18. Numerical evidence makes it plausible that there are infinitely many primes p such thatp+50 is also prime. List 15 of these primes.19. A positive integer n is called square-full, or powerful, if p2 |n for every prime factor pof n (there are 992 square-full numbers less than 250,000). If n is square-full, show thatit can be written in the form n = a²b³, with a and b positive integers.3.2 THE SIEVE OF ERATOSTHENESGiven a particular integer, how can we determine whether it is prime or compositeand, in the latter case, how can we actually find a nontrivial divisor? The mostobvious approach consists of successively dividing the integer in question by eachof the numbers preceding it; if none of them (except I) serves as a divisor, then theinteger must be prime. Although this method is very simple to describe, it cannotbe regarded as useful in practice. For even if one is undaunted by large calculationsthe amount of time and work involved may be prohibitive.There is a property of composite numbers that allows us to reduce materiallythe necessary computations-but still the process remains cumbersome. If an inthan 22 as nossible divisors, namely, the primes 2 3

Name: 15,16b 14. It has been conjectured that every even integer can be writ
Uploaded: 2024-03-20T06:46:03.000Z
Channel: Bartleby
Description: 15,16b 14. It has been conjectured that every even integer can be written as the difference of tb > 1, Theorem 3.2 ensures that b has at least one prime factor p. Then p 1 is composite, then it may be written as a = bc, where 1 1, it therefore suff