Let n be the smallest positive integer such that n(n+1)(n+2)(n+3) can be expressed as either a perfect square or a perfect cube (not necessarily both).

Find n, or prove that this is not possible.

let n(n+1)(n+2)(n+3)=d^2

or n(n+1)(n+2)(n+3)=c^3

thinking in terms of factors, the highest common factor n and (n+3) can have is 3. d^2 shld be a number with prime factors having even powers and c^3 shld be a number with prime factos having powers which are multiples of 3.

n=2 :2x3x4x5 is not a square nor a cube

n=3 :3x4x5x6 is not a square nor a cube

for n>3, n(n+1)(n+2)(n+3) would ALWAYS some prime factors which stand alone by themselves. As we know, a perfect square would have prime factors where the powers are even. since 3 is the highest common factor 2 of the terms in n, n+1, n+2, n+3 (a consecutive of 4 numbers) hence for n>3, n(n+1)(n+2)(n+3) has to have some other prime factors bigger than 3 and they appear only once.

hence showing that a perfect square or cube cannot exists.

in special cases where one of the terms in n,  n+1, n+2, n+3 is a perfect square, we can easily prove that although some prime factors larger than 3 may have powers which are even, the other three terms will DEFINITELY NOT be a perfect square, hence there still exists some prime factors which exist alone. same reasoning goes for cube.

 

Edited on February 25, 2004, 7:12 pm

Posted by tan on 2004-02-25 02:36:55