Show that if m is an integer then 3 divides m3−m.Show that the square of any integer is of the form 3m or 3m+1 but not of form 3m+2.Prove that the cube of any integer has to be exactly one of these forms: 9k, 9k + 1, or 9k + 8 for some integer k.Show that if a and b are relatively prime then (a+2b,2a+b)=1 or 3.Prove that the nth Fibonacci number fn≤αn−1, where α is the golden ratio.Show that if 2n−1 is prime then n is prime.Show that given any positive integer n, there exist n consecutive composite integers.Show that there are no prime triples other than 3, 5, 7.

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10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Show that if m is an integer then 3 divides m3−m.
Show that the square of any integer is of the form 3m or 3m+1 but not of form 3m+2.
Prove that the cube of any integer has to be exactly one of these forms: 9k, 9k + 1, or 9k + 8 for some integer k.
Show that if a and b are relatively prime then (a+2b,2a+b)=1 or 3.
Prove that the nth Fibonacci number fn≤αn−1, where α is the golden ratio.
Show that if 2n−1 is prime then n is prime.
Show that given any positive integer n, there exist n consecutive composite integers.
Show that there are no prime triples other than 3, 5, 7.