For any integer k greater than or equal to 2, define R(k) as the ratio of k 's largest positive prime divisor to its smallest positive prime divisor. For instance, R(14) = ½, R(15) = 3, R(30) = ½, and R(9) = 3 = 1. Note that R(k) = 1 if k is a prime number or any power of a prime. Additionally, R(1) is defined as 1. Prove or disprove these statements: (a) For every natural number x, there is a natural number y such that the ratio R(y) is at least x. (b) There is a natural number y such that for every natural number x, the ratio R(y) is at least x.
For any integer k greater than or equal to 2, define R(k) as the ratio of k 's largest positive prime divisor to its smallest positive prime divisor. For instance, R(14) = ½, R(15) = 3, R(30) = ½, and R(9) = 3 = 1. Note that R(k) = 1 if k is a prime number or any power of a prime. Additionally, R(1) is defined as 1. Prove or disprove these statements: (a) For every natural number x, there is a natural number y such that the ratio R(y) is at least x. (b) There is a natural number y such that for every natural number x, the ratio R(y) is at least x.
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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![For any integer k greater than or equal to 2, define R(k) as the ratio of k 's largest
positive prime divisor to its smallest positive prime divisor. For instance, R(14) = ½,
R(15) = 3, R(30) = ½, and R(9) = 3 = 1. Note that R(k) = 1 if k is a prime number or
any power of a prime. Additionally, R(1) is defined as 1.
Prove or disprove these statements: (a) For every natural number x, there is a natural
number y such that the ratio R(y) is at least x. (b) There is a natural number y such that
for every natural number x, the ratio R(y) is at least x.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3886e624-3ad7-4547-aeab-f36b3d1dd04a%2Faf9eb0d9-2f69-4dc5-bc32-1c5c9ce032bc%2Frr933ns_processed.png&w=3840&q=75)
Transcribed Image Text:For any integer k greater than or equal to 2, define R(k) as the ratio of k 's largest
positive prime divisor to its smallest positive prime divisor. For instance, R(14) = ½,
R(15) = 3, R(30) = ½, and R(9) = 3 = 1. Note that R(k) = 1 if k is a prime number or
any power of a prime. Additionally, R(1) is defined as 1.
Prove or disprove these statements: (a) For every natural number x, there is a natural
number y such that the ratio R(y) is at least x. (b) There is a natural number y such that
for every natural number x, the ratio R(y) is at least x.
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