4. Prove that every odd natural number is of one of the forms 4n+ 1 or 4n+3, where n is an integer. Your answer needs to be a little bit longer. Write a few sentences to complete your assignment. 5. Prove that for any integer n, at least one of the integers n, n+2, n + 4 is divisible by 3. 6. A classic unsolved problem in number theory asks if there are infinitely many pairs of "twin primes", pairs of primes separated by 2, such as 3 and 5, 11 and 13, or 71 and 73. Prove that the only prime triple (i.e. three primes, each 2 from the next) is 3, 5, 7. Your answer needs to be a little bit longer. Write a few sentences to complete your assignment. 7. Prove that for any natural number 11, 2+2²+2³+...+2=2+1-2

4. Prove that every odd natural number is of one of the forms 4n+ 1 or 4n+3, where n is an integer. Your answer needs to be a little bit longer. Write a few sentences to complete your assignment. 5. Prove that for any integer n, at least one of the integers n, n+2, n + 4 is divisible by 3. 6. A classic unsolved problem in number theory asks if there are infinitely many pairs of "twin primes", pairs of primes separated by 2, such as 3 and 5, 11 and 13, or 71 and 73. Prove that the only prime triple (i.e. three primes, each 2 from the next) is 3, 5, 7. Your answer needs to be a little bit longer. Write a few sentences to complete your assignment. 7. Prove that for any natural number 11, 2+2²+2³+...+2=2+1-2

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.2: Mathematical Induction

Problem 50E: Show that if the statement 1+2+3+...+n=n(n+1)2+2 is assumed to be true for n=k, the same equation...

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