Question 34722
not sure if this is a rigourous proof, but it should at least show you some things.


Take an example: 2*3*4*5. To be a perfect square, we need something like 3*3 or 7*7 or 15*15 etc.


So with 2*3*4*5, we need two of the integers to multiply to equal the other two integers. The only remote possibility of this is if 2*5=3*4, which it doesn't in this case, since 10 is not 12.


So, algebraically, x(x+1)(x+2)(x+3) are our four integers, starting at number "x".


If we then say, OK, for which value of x does x(x+3) = (x+1)(x+2) hold? then we can find the answer(s). From the wording of the question, we are looking for no solution.


{{{ x(x+3) = (x+1)(x+2) }}}
{{{ x^2+3x = x^2 + 3x + 2 }}}
{{{ 0 = 2 }}}
which is patently not true, so there is no value of x such that {{{ x(x+3) = (x+1)(x+2) }}} is true.


jon.