Question

We can see that 3*4*5*6 = 360 = 19^2-1.
Prove that the product of four consecutive integers is always one less than a perfect
square.

Answer 1

(x)(x+1)(x+2)(x+3)=6 x + 11 x^2 + 6 x^3 + x^4
6 x + 11 x^2 + 6 x^3 + x^4=a^2 - 1
a=\[\sqrt{1+6 x+11 x^2+6 x^3+x^4}\]
a=\[\left(1+3 x+x^2\right)\]

Answer 2

2 x 3 x4 x 5=120

Answer 3

This is really a not good proof but it does find what number have to be squared.

Answer 4

why dont you like it?

Answer 5

11^2-1=120

Answer 6

2*3*4*5=120

which is 121(perfect square -1)

Answer 7

Ah so, maybe you come up with something good lol

Answer 8

because it only find a, not really a definite proof

Answer 9

Yeah, some math magic.