Question

Show that any positive number of the form 4n-1 is divisible by a prime of the...

Show that any positive number of the form 4n-1 is divisible by a prime of the same form.

Homework Answers

Answer #1

We know that there are primes of the form 2, or of the form

Primes of the form don't exist (except 2 which is a prime of the form and the only one of this sort)

Let be any number and let its prime factorization be where are all primes

If none of these primes are of the form they must either be 2 or of the form

Note that

If one of them is 2 then for

This means can possibility only be

But is modulo 4

So this contradicts our assumption that none of the primes in the prime factorization are of the form

So that every number of the form must have a prime factor of the same form

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
11n-4n is divisible by 7 for any n >0
11n-4n is divisible by 7 for any n >0
What is the smallest non-prime natural number that is not divisible by any number on the...
What is the smallest non-prime natural number that is not divisible by any number on the following list? 2, 3, 5, 7, 11, 13, 17. Explain your reasoning.
A prime number is an integer greater than 1 that is evenly divisible by only 1...
A prime number is an integer greater than 1 that is evenly divisible by only 1 and itself. For example, 2, 3, 5, and 7 are prime numbers, but 4, 6, 8, and 9 are not. Create a PrimeNumber application that prompts the user for a number and then displays a message indicating whether the number is prime or not. Hint: The % operator can be used to determine if one number is evenly divisible by another. ( Java programing...
Let m,n be any positive integers. Show that if m,n have no common prime divisor (i.e....
Let m,n be any positive integers. Show that if m,n have no common prime divisor (i.e. a divisor that is at the same time a prime number), then m+n and m have no common prime divisor. (Hint: try it indirectly)
Use a proof by induction to show that, −(16−11?) is a positive number that is divisible...
Use a proof by induction to show that, −(16−11?) is a positive number that is divisible by 5 when ? ≥ 2. Prove (using a formal proof technique) that any sequence that begins with the first four integers 12, 6, 4, is neither arithmetic nor geometric.
Show that, for any integer n ≥ 2, (n + 1)n − 1 is divisible by...
Show that, for any integer n ≥ 2, (n + 1)n − 1 is divisible by n2 . (Hint: Use the Binomial Theorem.)
Let n be a positive integer. Show that every abelian group of order n is cyclic...
Let n be a positive integer. Show that every abelian group of order n is cyclic if and only if n is not divisible by the square of any prime.
1. Let p be any prime number. Let r be any integer such that 0 <...
1. Let p be any prime number. Let r be any integer such that 0 < r < p−1. Show that there exists a number q such that rq = 1(mod p) 2. Let p1 and p2 be two distinct prime numbers. Let r1 and r2 be such that 0 < r1 < p1 and 0 < r2 < p2. Show that there exists a number x such that x = r1(mod p1)andx = r2(mod p2). 8. Suppose we roll...
Prove the following statements: 1- If m and n are relatively prime, then for any x...
Prove the following statements: 1- If m and n are relatively prime, then for any x belongs, Z there are integers a; b such that x = am + bn 2- For every n belongs N, the number (n^3 + 2) is not divisible by 4.
Use the method of direct proof to show that for any positive 5-digit integer n, if...
Use the method of direct proof to show that for any positive 5-digit integer n, if n is divisible by 9, then some of its digits is divisible by 9 too.