Could someone help me with this induction proof. I know its true.

given then any integer m is less 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

I made a mistake... i meant any integer m is greater than or equal to 2

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 the base case for m = 2:
For m = 2, we need to find a sequence of (2-1) = 1 positive integer which is not prime. The only positive integer satisfying this condition is 1, which is not prime. So, the base case holds.

Now, let's assume that for some positive integer k ≥ 2, there exists a sequence of (k-1) consecutive positive integers, none of which is prime. This is our induction hypothesis.

We need to prove that there also exists a sequence of (k+1)-1 = k consecutive positive integers, none of which is prime.

Consider the sequence of (k-1) consecutive positive integers assumed in our hypothesis: n, n+1, n+2, ..., n+(k-2). None of these integers is prime.

To prove that there exists a sequence of k consecutive positive integers, none of which is prime, we just need to consider the next integer after n+(k-2), which is n+(k-2)+1. We claim that this integer is composite, meaning it is not prime.

To see why this claim is true, consider the possible factors of n+(k-2)+1:
1. It is not divisible by 2, since it is odd (as k ≥ 2).
2. It is not divisible by 3, since the sum of the digits is not divisible by 3 (as k ≥ 2).
3. It is not divisible by 5, since it does not end in 0 or 5 (as k ≥ 2).
4. It is not divisible by any integer greater than 5 and less than or equal to √(n+(k-2)+1), because we already assumed all the integers up to n+(k-2) are not prime.

Since the integer n+(k-2)+1 is not divisible by any prime number less than or equal to √(n+(k-2)+1), it must be composite (i.e., not prime).

So, we have shown that if our induction hypothesis holds for k, then it also holds for k+1.

By the principle of mathematical induction, we conclude 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.