how do you prove that the product of four consecutive integers is one less than a perfect square. I started by letting m be the lowest so m(m+1)(m+2)(m+3) + 1 should be a perfect square, i don't know how to show this algebraically :(

Printable View

- May 16th 2008, 07:26 PMeniuqvwproof about product of consecutive ints
how do you prove that the product of four consecutive integers is one less than a perfect square. I started by letting m be the lowest so m(m+1)(m+2)(m+3) + 1 should be a perfect square, i don't know how to show this algebraically :(

- May 16th 2008, 08:51 PMReckoner
Hi, eniuqvw!

Obviously, all that is needed is to show that can be factored into the square of a quadratic in . If you can factor this expression right away, then good for you! I, however, didn't have much luck (if anyone has a nice and simple way to factor this, let me know!), so I worked backwards:

The square of a quadratic will have the form

So, if can indeed be factored as the square of a quadratic in , then it should take that form. With this assumption, we have:

Solving this, we find that .

And, indeed, if you try expanding it yourself, you will see that

And this fact should give you what you need to complete your proof. - May 16th 2008, 10:08 PMSoroban
Hello, eniuqvw!

Quote:

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

We have: .

. .

. . . .

. . . .

. . . .