Can a Sequence of Consecutive Positive Integers Not Contain Any Primes?

In summary, the conversation is about an induction proof where the question is whether it is possible to find a sequence of m-1 consecutive positive integers, none of which are prime, given an integer m greater than or equal to 2. The speaker named ZioX is looking for help with this proof and mentions that their teacher only gave them true proofs to prevent them from finding counter examples. Another speaker, kai89, suggests using divisibility to solve the problem. ZioX then clarifies their initial statement and explains how using divisibility can lead to finding a sequence of consecutive non-prime integers. The conversation ends with kai89 complimenting ZioX's solution and joking about the unpredictability of prime numbers.
  • #1
kai89
4
0
Could someone help me with this induction proof. I know its true.

given 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
 
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  • #2
Is 5! prime?
 
  • #3
I'm sorry, Ziox, I really don't see what that has to do with the problem. Please enlighten me.
 
  • #4
How do you know this is true?
 
  • #5
My teacher only gave us true proofs so we wouldn't be able to prove it wrong by counter example.
 
  • #6
HallsofIvy said:
I'm sorry, Ziox, I really don't see what that has to do with the problem. Please enlighten me.

5!+i has divisors 2,3,4,5 for i=2,...,5. Paired up respectively.
 
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  • #7
Oh, of course! That answers the whole question. I was focusing on the "5" and didn't think about doing the same thing for n in general. Very nice.

kai89, do you understand what ZioX is saying?
 
  • #8
HallsofIvy said:
Oh, of course! That answers the whole question. I was focusing on the "5" and didn't think about doing the same thing for n in general. Very nice.

kai89, do you understand what ZioX is saying?

You're right though, I was being pretty vague and probably would've only made sense if someone has seen it before. Should've said something about divisibility when adding.
 
  • #9
kai89, since it has been a couple of days now, I will give detail on what ZioX hinted at: For any positive integer n, n! is obviously divisible by every integer up to and including n. Therefore, n!+ 2 is divisible by 2, n!+ 3 is divisible by 3, up to n!+ n is divisible by by n. You have n-1 consecutive integers that are not prime.

As I said before, very nice!
 
  • #10
Primes are so unpredictable.

;-p
 

1. What is discrete math induction proof?

Discrete math induction proof is a mathematical method used to prove statements about integers or other discrete objects. It involves using a base case and an inductive step to show that a statement holds for all integers or objects in a given set.

2. How does discrete math induction proof differ from other proof techniques?

Unlike other proof techniques that involve direct or indirect proof, discrete math induction proof relies on the principle of mathematical induction to prove a statement. This principle states that if a statement holds for a base case and can be proven to hold for the next case, then it holds for all subsequent cases.

3. What types of statements can be proven using discrete math induction proof?

Discrete math induction proof is commonly used to prove statements about integers, such as properties of sequences, series, and sets. It can also be used to prove statements about other discrete objects, such as graphs, trees, and proofs by contradiction.

4. What is the process for conducting a discrete math induction proof?

The process for conducting a discrete math induction proof involves three steps: 1) establishing a base case, which proves the statement for the first integer or object in the set, 2) assuming the statement holds for a general case (usually denoted by k), and 3) using the assumption to prove the statement holds for the next case (k+1). By repeating this process, the statement can be proven to hold for all integers or objects in the set.

5. Are there any tips for successfully completing a discrete math induction proof?

Some tips for successfully completing a discrete math induction proof include: 1) carefully choosing the base case, 2) clearly stating the inductive hypothesis (the assumption for the general case), 3) using precise mathematical language and notation, and 4) providing a clear and logical proof for the inductive step. It is also important to practice and familiarize oneself with various examples of discrete math induction proof to build proficiency in this technique.

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