can anyone show me the proof that "the product of four positive integers in arithmetic progression cannot be the square of an integer".
thanks!
Hello, sheryl!
I know a proof for a similar problem, but I've hit a wall.
Let me show you the groundwork . . . maybe you can finish it.
The product of four positive integers in arithmetic progression
cannot be the square of an integer.
Let the four numbers be: .$\displaystyle a,\;a+d,\;a+2d,\;a+3d$
Suppose their product is a square.
Then we have: .$\displaystyle a(a+d)(a+2d)(a+3d)\;=\;k^2$
Regroup: .$\displaystyle [a(a+3d)]\,[(a+d)(a+2d)] \;=\;k^2 $
Multiply: .$\displaystyle [a^2+3ad)\,(a^2+3ad + 2d^2)\;=\;k^2$
And we have: .$\displaystyle [(a^2 + 3ad + d^2) - d^2]\,[(a^2+3ad + d^2) + d^2] \;= \;k^2$
Multiply: .$\displaystyle (a^2+3ad + d^2)^2 - d^4 \;= \;k^2$
And this is supposed to lead to a contradiction . . . but I can't find it.
Anyone? .Anyone?
Hello, all!
The problem I referred to is:
"Prove that the product of four consecutive positive integers cannot be a square."
Let the integers be: $\displaystyle a,\;a+1,\;a+2,\;a+3$
Suppose their product is a square.
Then we have: .$\displaystyle a(a+1)(a+2)(a+3) \:=\:k^2$
Regrouping: .$\displaystyle \left[a(a+3)\right]\,\left[(a+1)(a+2)\right] \:=\:k^2$
We have: .$\displaystyle [a^2 + 3a]\,[a^2+3a +2] \:=\:k^2$
. . then: .$\displaystyle \left[(a^2 + 3a + 1) - 1\right]\,\keft[(a^2 + 3a + 1) + 1\right] \:= \:k^2$
Hence: .$\displaystyle (a^2 + 3a + 1)^2 - 1^2\:=\:k^2\quad\Rightarrow\quad (a^2 + 3a + 1)^2 - k^2\:=\:1$
We have: "The difference of two squares is $\displaystyle 1.$"
But this is true for $\displaystyle (\pm1)^2 - 0^2$ only.
Then: .$\displaystyle a^2 + 3a + 1 \:= \:\pm1\quad\Rightarrow\quad\left\{\begin{array}{cc }a(a +3) \:= \:0 \\ (a+1)(a+2) \:= \:0\end{array}$
. . and $\displaystyle a$ is not a positive integer.
It is just enthusiasm you talked about/Originally Posted by Soroban
Let gcd(a,d)=c
Let a=cm, d=cn
Then your expression will be c^4 m(m+n)(m+2n)(m+3n)
Now, since m and n are relatively prime, any two factors are relatively prime to each other(may have common factor of 2 or 3)
Hence their product cannot be a cannot be a perfect square.
Keep Smiling
Malay