(b) Every integer n> I is the product of a square-free integer and a perfect square. [Hint: If n = p' p •· p is the canonical factorization of n, then write k, = 24, +r, where r =0 or 1 according as k, is even or odd.] Verify that any integer n can be %3! ...
(b) Every integer n> I is the product of a square-free integer and a perfect square. [Hint: If n = p' p •· p is the canonical factorization of n, then write k, = 24, +r, where r =0 or 1 according as k, is even or odd.] Verify that any integer n can be %3! ...
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
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16b
![79% D
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
b > 1, Theorem 3.2 ensures that b has at least one prime factor p. Then p<bs ya;
furthermore, because p|b and b|a, it follows that p|a. The point is simply this: a
Inasmuch as 22 < 509 < 23, we need only try out the primes that are no
list of primes up to Ja). This may be clarified by considering the integer a=509.
a by those primes not exceeding Ja (presuming, of course, the availability of a
composite number a will always possess a prime divisor p satisfying p < Vā.
ll Pure Talk
2:39 PM
14. It has been conjectured that every even integer can be written as the difference of t
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 = p p .· p is the canonical factorization of n, then write k, =
291 +rị where ri = 0 or 1 according as k; is even or odd.]
17. Verify that any integer n can be expressed asn=2*m, 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 1) 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
teger a > 1 is composite, then it may be wrtten as a = bc, where 1 < b < a and
p.
han 22 as possible divisors, namely, the primes 2 35.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7dbb4ae4-0d65-4baa-9481-63f79be91eca%2F82af2375-65b7-4a71-b892-892f6ad22a5b%2Fxk1lbta_processed.jpeg&w=3840&q=75)
Transcribed Image Text:79% D
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
b > 1, Theorem 3.2 ensures that b has at least one prime factor p. Then p<bs ya;
furthermore, because p|b and b|a, it follows that p|a. The point is simply this: a
Inasmuch as 22 < 509 < 23, we need only try out the primes that are no
list of primes up to Ja). This may be clarified by considering the integer a=509.
a by those primes not exceeding Ja (presuming, of course, the availability of a
composite number a will always possess a prime divisor p satisfying p < Vā.
ll Pure Talk
2:39 PM
14. It has been conjectured that every even integer can be written as the difference of t
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 = p p .· p is the canonical factorization of n, then write k, =
291 +rị where ri = 0 or 1 according as k; is even or odd.]
17. Verify that any integer n can be expressed asn=2*m, 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 1) 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
teger a > 1 is composite, then it may be wrtten as a = bc, where 1 < b < a and
p.
han 22 as possible divisors, namely, the primes 2 35.
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