Proof about product of 4 integers

In summary, the product of four consecutive integers can be expressed as (n-1)(n)(n+1)(n+2), and by factoring this into (n^2-1)(n)(n+2), we can see that it is always one less than a perfect square. By taking different pairs of factors or factorizing the expression (n-1)(n)(n+1)(n+2) + 1, we can easily prove this statement.
  • #1
cragar
2,552
3

Homework Statement


Prove that the product of four consecutive integers is always one less than a perfect square.

The Attempt at a Solution


I tried looking at the product [itex] (n-1)(n)(n+1)(n+2)=x^2-1 [/itex]
but i couldn't seem to get anything useful out of it. I added one to both sides .
I tried to see if i could some how show that the product of the four integers plus 1 had an odd number of divisors but I couldn't see a way to do it. I did notice that these numbers are never divisible by 2,3,or 4, and a lot of the time they produce a prime, all though I didn't check very many.
And when it wasn't a prime it was divisible by 5. I was trying to think of a way to do this with Pythagorean triples but I am not sure.
 
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  • #2
cragar said:
I tried looking at the product [itex] (n-1)(n)(n+1)(n+2)=x^2-1 [/itex]
but i couldn't seem to get anything useful out of it.

Look harder :smile:. Don't overthink the problem with the other ideas you tried. This doesn't depend on any theorems in number theory!

It's hard to give a hint without telling you the answer, but think about what you can you do with ##x^2-1##.
 
  • #3
are you talking about factoring it into (x+1)(x-1) I not sure how that helps I did look at that. we do know that x needs to be odd. so x=2y+1 so (2y+1+1)(2y+1-1)=(2y+2)(2y)=4y(y+1) I still don't see how it will work. I guess Ill keep trying stuff. thanks for your post.
 
  • #4
cragar said:
are you talking about factoring it into (x+1)(x-1) I not sure how that helps

Don't bother about x being odd or even. Just factor the left hand side into the same form.
 
  • #5
So we have (n^2-1)(n)(n+2)=x^2-1 I just don't see what this does, I can see how we have similar things on both sides. but it seems like we still have 2 extra terms on the left side.
 
  • #6
Try taking different pairs of factors from ##(n-1)(n)(n+1)(n+2)## to get ##(an^2 + bn + c -1)(an^2 + bn + c + 1)##.

Or, factorize ##(n-1)(n)(n+1)(n+2) + 1## into a perfect square, like ##(an^2 + bn + c)^2##. It's not hard to guess the values of ##a## and ##c##.
 
  • #7
ok thanks for your help, I see it now
 

Related to Proof about product of 4 integers

1. What is a proof about the product of 4 integers?

A proof about the product of 4 integers is a mathematical demonstration that shows the product of any 4 whole numbers will always yield a whole number result.

2. Why is a proof about the product of 4 integers important?

A proof about the product of 4 integers is important because it helps to solidify the understanding of basic mathematical concepts, such as multiplication and whole numbers. It also serves as a foundation for more complex mathematical proofs and theories.

3. How is a proof about the product of 4 integers different from other mathematical proofs?

A proof about the product of 4 integers is specific to the properties and rules of multiplication, while other mathematical proofs may focus on different operations or concepts. Additionally, a proof about the product of 4 integers often involves the use of algebraic equations and properties, rather than just logical arguments.

4. Are there any real-world applications for a proof about the product of 4 integers?

Yes, there are many real-world applications for a proof about the product of 4 integers. For example, it can be used in engineering and construction to ensure accurate measurements and calculations, or in computer programming to optimize algorithms and data structures.

5. How can I understand and replicate a proof about the product of 4 integers?

To understand and replicate a proof about the product of 4 integers, it is important to have a strong understanding of basic mathematical concepts and properties, such as commutativity and distributivity. It may also be helpful to study and analyze other mathematical proofs to gain a better understanding of the logic and techniques used.

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