18 thermore, because p|b and b|a, it follows that p|a. The point is simply this: a44ELEMENTARY NUMBER THEORYconsecutive 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 esall 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 integern> 1 is square-free if and only if n can be factored into a product aedistinct primes.(b) Every integer n > 1 is the product of a square-free integer and a perfect square.[Hint: If n = p p29; +rị where r; = 0 or 1 according as k; is even or odd.]... pk is the canonical factorization of n, then write k: -2km, wherek>0 and m is an odd17. Verify that any integer n can be expressed as n =integer.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 = a²b°, 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 1) serves as a divisor, then thenteger must be prime. Although this methode regarded as useful in practice. For even if one is undaunted by large calculations.ne amount of time and work involved may be prohibitive.There is a property of composite numbers that allows us to reduce materiallye necessary computations-but still the process remains cumbersome. If an inger a > 1 is composite, then it may be written as a =

Name: 18 thermore, because p|b and b|a, it follows that p|a. The point is si
Uploaded: 2024-03-20T07:06:41.000Z
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Description: 18 thermore, because p|b and b|a, it follows that p|a. The point is simply this: a44ELEMENTARY NUMBER THEORYconsecutive primes in infinitely many ways. For example,6 = 29- 23 = 137 – 131 = 599 – 593 = 1019 – 1013 = . ..Express the integer 10 as the d