   SEARCH HOME Math Central Quandaries & Queries  Question from kim, a parent: why can't a square number double another one Do you mean why can't a square be twice another square?

Any integer can be broken down into a power of 2 times its 'odd part'. For example, 60 = 22 x 15, 1000 = 23 x 125, 37 =20 x 37, 38 = 21 x 19, etc. Thus if a square was twice another square, say m2 = 2xn2 let's think about the odd part of the numbers on each side of the equation. They must be the same and we can cancel them leaving just powers of 2 on both sides of the equation. Now comes a little bit of a subtle part. If we square an even number when we look at the result and break it into its odd part and powers of 2 the number of powers of two must be even; for example if we square 60 then the result is 24 x 152, if we square 37 we end up with 20 x 372, if we square 1000 we end up with 26 x 1252, etc. Thus if we had m2 = 2 x n2 the number of powers of 2 in m2 is even, likewise the number of powers of 2 in n2 is even but then we would have an even number of powers of 2 on the left hand side of the equation but (because of the extra 2) an odd number of powers of 2 on the right hand side of the equation and this can't be so as they're equal. Thus a square can't be twice another square

Hope this helps,

Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.