15,16b 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...

15,16b

14. It has been conjectured that every even integer can be written as the difference of t<br>b > 1, Theorem 3.2 ensures that b has at least one prime factor p. Then p <b< Va;<br>furthermore, because p|b and b|a, it follows that p|a. The point is simply this: a<br>list of primes up to Va). This may be clarified by considering the integer a = 509.<br>a by those primes not exceeding Ja (presuming, of course, the availability of a<br>teger a > 1 is composite, then it may be written as a = bc, where 1 <b < a and<br>composite number a will always possess a prime divisor p satisfying p<Va.<br>Inasmuch as 22 < 509 < 23, we need only try out the primes that are not<br>In testing the primality of a specific integer a > 1, it therefore suffices to divide<br>1 <c < a. Assuming that b<c, we get b2 < bc = a, and so b < Va. Because<br>consecutive primes in infinitely many ways. For example,<br>6 = 29 - 23 = 137-131 = 599 - 593 1019-1013 =..<br>Express the integer 10 as the difference of two consecutive primes in 15 ways.<br>15. Prove that a positive integer a > 1 is a square if and only if in the canonical form of a<br>all the exponents of the primes are even integers.<br>16. An integer is said to be square-free if it is not divisible by the square of any integer greater<br>than 1. Prove the following:<br>(a) An integer n> 1 is square-free if and only if n can be factored into a product of<br>distinct primes.<br>(b) Every integer n > 1 is the product of a square-free integer and a perfect square.<br>[Hint: If n =<br>29; +ri where r¡ = 0 or 1 according as k; is even or odd.]<br>ki „k2<br>PP p is the canonical factorization of n, then write k; =<br>17. Verify that any integer n can be expressed as n = 2km, where k >0 and m is an odd<br>integer.<br>18. Numerical evidence makes it plausible that there are infinitely many primes p such that<br>p+50 is also prime. List 15 of these primes.<br>19. A positive integer n is called square-full, or powerful, if p2 |n for every prime factor p<br>of n (there are 992 square-full numbers less than 250,000). If n is square-full, show that<br>it can be written in the form n = a²b³, with a and b positive integers.<br>3.2 THE SIEVE OF ERATOSTHENES<br>Given a particular integer, how can we determine whether it is prime or composite<br>and, in the latter case, how can we actually find a nontrivial divisor? The most<br>obvious approach consists of successively dividing the integer in question by each<br>of the numbers preceding it; if none of them (except I) serves as a divisor, then the<br>integer must be prime. Although this method is very simple to describe, it cannot<br>be regarded as useful in practice. For even if one is undaunted by large calculations<br>the amount of time and work involved may be prohibitive.<br>There is a property of composite numbers that allows us to reduce materially<br>the necessary computations-but still the process remains cumbersome. If an in<br>than 22 as nossible divisors, namely, the primes 2 3<br>

Extracted text: 14. It has been conjectured that every even integer can be written as the difference of t b > 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 divide 1 < a.="" assuming="" that="">< bc="a," and="" so="" b="">< va.="" because="" consecutive="" primes="" in="" infinitely="" many="" ways.="" for="" example,="" 6="29" -="" 23="137-131" =="" 599="" -="" 593="" 1019-1013=".." express="" the="" integer="" 10="" as="" the="" difference="" of="" two="" consecutive="" primes="" in="" 15="" ways.="" 15.="" prove="" that="" a="" positive="" integer="" a=""> 1 is a square if and only if in the canonical form of a all 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 greater than 1. Prove the following: (a) An integer n> 1 is square-free if and only if n can be factored into a product of distinct 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 „k2 PP 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 odd integer. 18. Numerical evidence makes it plausible that there are infinitely many primes p such that p+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 p of n (there are 992 square-full numbers less than 250,000). If n is square-full, show that it can be written in the form n = a²b³, with a and b positive integers. 3.2 THE SIEVE OF ERATOSTHENES Given a particular integer, how can we determine whether it is prime or composite and, in the latter case, how can we actually find a nontrivial divisor? The most obvious approach consists of successively dividing the integer in question by each of the numbers preceding it; if none of them (except I) serves as a divisor, then the integer must be prime. Although this method is very simple to describe, it cannot be regarded as useful in practice. For even if one is undaunted by large calculations the amount of time and work involved may be prohibitive. There is a property of composite numbers that allows us to reduce materially the necessary computations-but still the process remains cumbersome. If an in than 22 as nossible divisors, namely, the primes 2 3

Jun 03, 2022

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15,16b 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...

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