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: .
Suppose their product is a square.
Then we have: .
Regroup: .
Multiply: .
And we have: .
Multiply: .
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:
Suppose their product is a square.
Then we have: .
Regrouping: .
We have: .
. . then: .
Hence: .
We have: "The difference of two squares is "
But this is true for only.
Then: .
. . and 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