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
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
Hi, eniuqvw!
Obviously, all that is needed is to show that $\displaystyle m\left(m+1\right)\left(m+2\right)\left(m+3\right)+ 1 =
m^4+6m^3+11m^2+6m+1$ can be factored into the square of a quadratic in $\displaystyle m$. 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
$\displaystyle \left(am^2 + bm + c\right)^2 =
a^2m^4+2abm^3+\left(2ac+b^2\right)m^2+2bcm+c^2$
So, if $\displaystyle m^4+6m^3+11m^2+6m+1$ can indeed be factored as the square of a quadratic in $\displaystyle m$, then it should take that form. With this assumption, we have:
$\displaystyle \begin{array}{rcl}
a^2 & = & 1\\
2ab & = & 6\\
2ac + b^2 & = & 11\\
2bc & = & 6\\
c^2 & = & 1
\end{array}$
Solving this, we find that $\displaystyle \left(a,\,b,\,c\right) = \left(1,\,3,\,1\right)\text{ or }\left(-1,\,-3\,-1\right)$.
And, indeed, if you try expanding it yourself, you will see that
$\displaystyle \left(m^2 + 3m + 1\right)^2 = m^4+6m^3+11m^2+6m+1$
And this fact should give you what you need to complete your proof.
Hello, eniuqvw!
Let the four integers are: .$\displaystyle m,\:m+1,\:m+2,\:m+3$Prove that the product of four consecutive integers is one less than a perfect square.
We have: .$\displaystyle P \;=\;m(m+1)(m+2)(m+3)$
. . $\displaystyle P \;=\;[m(m+3)]\,[(m+1)(m+2)]$
. . . . $\displaystyle =\;[m^2 + 3m]\,[m^2 + 3m + 2]
$
. . . . $\displaystyle = \;[(m^2+3m + 1) - 1]\,[(m^2+3m + 1) + 1] $
. . . . $\displaystyle =\;(m^2 + 3m + 1)^2 - 1^2$