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.
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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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Section: Chapter Questions
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15,16b
![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 <b< Va;
furthermore, because p|b and b|a, it follows that p|a. The point is simply this: a
list of primes up to Va). This may be clarified by considering the integer a = 509.
a by those primes not exceeding Ja (presuming, of course, the availability of a
teger a > 1 is composite, then it may be written as a = bc, where 1 <b < a and
composite number a will always possess a prime divisor p satisfying p<Va.
Inasmuch as 22 < 509 < 23, we need only try out the primes that are not
In testing the primality of a specific integer a > 1, it therefore suffices to divide
1 <c < a. Assuming that b<c, we get b2 < 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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7dbb4ae4-0d65-4baa-9481-63f79be91eca%2F70105074-41c9-4d59-9310-85bc250bbffd%2F1zrs2xq_processed.jpeg&w=3840&q=75)
Transcribed Image 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 <b< Va;
furthermore, because p|b and b|a, it follows that p|a. The point is simply this: a
list of primes up to Va). This may be clarified by considering the integer a = 509.
a by those primes not exceeding Ja (presuming, of course, the availability of a
teger a > 1 is composite, then it may be written as a = bc, where 1 <b < a and
composite number a will always possess a prime divisor p satisfying p<Va.
Inasmuch as 22 < 509 < 23, we need only try out the primes that are not
In testing the primality of a specific integer a > 1, it therefore suffices to divide
1 <c < a. Assuming that b<c, we get b2 < 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
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