Fun with mathematics

Creative thought-

From   R ^2   to   R ^2 – relation

          2  1           2  2

The Pythagorean triples are having its own importance in

recreational mathematics. It is a source of generating

higher order like power numeral relations,particularly

with squares. In fact there is a close relationship between 

   R ^2   and   R ^2 –relations

2   1            2   2

In   R ^2 . a square is equated with a sum of two squares  and in 

    2   1

 R ^2  , a sum of two squares is equated with a sum of two other

2  2     

squares. By using a Pythagorean triple or two sets of triples,

one can generate     R^2

        2  2

 With one Pythagorean triple (x,y,z) we can generate

(x+z)^2 + (y+z)^2 = (z+x+y)^2 + (y-x)^2

For example (3,4,5) yields,

8^2 + 9^2 = 12^2 + 1^2

It is curious to  know that a single R^2 relation can generate
                                                                     2  1

a series of R^2 relations . For example the above numeral

relation can be written with numbers existing and number

added

(3+5)^2 + (4+5)^2 = (5+7)^2 + (0+1)^2

If all the existing numbers only or if  all the added numbers

only are multiplied with a number n , the equality of the relation

is not affected.

 (3n+5)^2 + (4n+5)^2 = (5n+7)^2 + 1^2

or

(3+5n)^2 +(4+5n)^2 = (5+7n)^2 + n^2

Again if all the existing numbers are multiplied with m and

all added numbers with n, the equality of the relation n is

still preserved.

(3m+5n)^2 + (4m+5n)^2 = (5m+7n)^2 + n^2