



 
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 = 2^{2} x 15, 1000 = 2^{3} x 125, 37 =2^{0 }x 37, 38 = 2^{1 }x 19, etc. Thus if a square was twice another square, say m^{2} = 2xn^{2} 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 2^{4} x 15^{2}, if we square 37 we end up with 2^{0} x 37^{2}, if we square 1000 we end up with 2^{6} x 125^{2}, etc. Thus if we had m^{2} = 2 x n^{2} the number of powers of 2 in m^{2} is even, likewise the number of powers of 2 in n^{2} 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  


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