0

Can a square number be split as the sum of two other squares in two different ways? The Pythagorean numbers help us to identify numbers which can be split as sum of two squares. Is this unique?

Narayanan Raman
  • 1

3 Answers3

3

By the Brahmagupta–Fibonacci identity, which states that $$(a^2 + b^2)(c^2 + d^2) = (ac-bd)^2 + (ad+bc)^2 = (ac+bd)^2 + (ad-bc)^2$$ you can get a number which is a sum of two squares two different ways by multiplying two sums of two squares.

So from two Pythagorean triples $a^2 + b^2 = a_1^2$ and $c^2 + d^2 = a_2^2$ , the left hand side simplifies to $(a_1a_2)^2$, you can get a number that can be written as a sum of two squares.

twnly
  • 1,700
  • Love whipping out the Brahmagupta-Fibonacci identity. https://math.stackexchange.com/questions/1012733/prove-that-a-positive-polynomial-function-can-be-written-as-the-squares-of-two-p/1012765#1012765 – Matt Samuel Dec 17 '18 at 02:40
  • 1
    Of course if you have $a^2 + b^2 = a_1^2$ and $c^2 + d^2 = a_2^2$ then you also have $(a_2a)^2 + (a_2b)^2 = (a_1a_2)^2$ and $(a_1c)^2 + (a_1d)^2 = (a_1a_2)^2.$ This is how $65^2$ comes to be the sum of two squares in four different ways: take $a=4,b=3,c=12,d=5.$ – David K Dec 17 '18 at 12:00
0

HINT

$3,4,5$ is a PPT, meaning that any multiple (like $6,8,10$) is a PT

$5,12,13$ is a PPT as well ...

Bram28
  • 100,612
  • 6
  • 70
  • 118
0

For example \begin{eqnarray*} 65^2=63^2+16^2=33^2+56^2. \end{eqnarray*}

Donald Splutterwit
  • 36,613
  • 2
  • 26
  • 73