Could someone help me with this induction proof. I know its true.
given then any integer m is greater than or equal to 2, is it possible to find a sequence of m-1 consecutive positive integers none of which is prime? explain
any help is greatly appreciated thanks
Does the sequence have to start at m?
To prove that for any integer m greater than or equal to 2, there exists a sequence of m-1 consecutive positive integers, none of which is prime, we can use mathematical induction.
First, let's establish a base case. When m = 2, we need to find a sequence of 2-1 = 1 positive integer that is not prime. The only positive integer less than or equal to 1 is 1 itself, which is not prime. Therefore, the base case is true.
Now, let's assume that for some integer k ≥ 2, there exists a sequence of k-1 consecutive positive integers, none of which is prime.
To prove that the statement holds for k+1, we need to find a sequence of (k+1)-1 = k consecutive positive integers, none of which is prime.
For the sequence to start at k+1, we would have k+1, k+2, k+3, ..., 2k. However, this sequence may contain prime numbers. So, we need to consider a different approach.
Let's start the sequence at m = 2. The sequence would then be 2, 3, 4, ..., k+1. The length of this sequence is k, which matches the requirement.
Now, we need to show that none of the integers in this sequence are prime. Since 2 is the first element, which is not prime, we only need to focus on the remaining numbers.
We know that 1 is not prime, so we can exclude it from consideration. As for the rest of the numbers from 3 to k+1, each of these numbers can be written as a product of two factors, neither of which equals 1 or the number itself. For example:
3 = 1 * 3
4 = 2 * 2
5 = 1 * 5
...
k+1 = 1 * (k+1)
Therefore, each number can be expressed as a product of integers greater than 1 and less than itself. This means that no number within this sequence is prime.
By proving the base case and showing that if the statement holds for an integer k, it also holds for k+1, we have established the statement for all integers greater than or equal to 2 using mathematical induction.
Hence, for any integer m ≥ 2, it is possible to find a sequence of m-1 consecutive positive integers, none of which is prime.