how to find three positive integers $\mathbf a, \mathbf b, \mathbf a+\mathbf b$ each can be write as a sum of two squares

Question

I need to find $\mathbf a, \mathbf b \in \Bbb Z^+$

such that $\mathbf a$ can be write as a sum of two squares

and $\mathbf b$ can be write as a sum of two squares

and $\mathbf a+\mathbf b$ can be write as a sum of two squares

is there are any ?

i tried to find but i can't thanks.

Answer 1

The OP should have worked more. If $0$ is counted as a square, then the smallest examples (i.e., smallest $a+b$) are $\{a,b\}=\{1,1\}$ if $a=b$ is possible, and $\{a,b\}=\{1,4\}$ if $a\neq b$ must hold. If $0$ is not included, then the smallest examples (i.e., smallest $a+b$) are $\{a,b\}=\{5,5\}$ if $a=b$ is possible, and $\{a,b\}=\{5,8\}$ if $a\neq b$ must hold. One does not need to know any positive integer above $13$ to answer this question.

Answer 2

$$a=3^2+4^2=25$$ $$b=6^2+8^2=100$$

$$a+b=125=10^2+5^2$$

The key to generating such pairs is to have a look at Pythagorean triples

Answer 3

A positive integer $n$ is representable as the sum of two squares as $n=a^2+b^2$ only if every prime divisor $p\equiv3 \mbox{ mod }4$ of $n$ occurs with an even exponent.

Answer 4

For the equation:

$$X_1^2+X_2^2=Y_1^2+Y_2^2+Y_3^2+Y_4^2$$

Solutions have the form:

$$X_1=t^2+2(p+s-y)t+k^2+2y^2+2p^2-4yp-2ys+4ps$$

$$X_2=t^2+2(p+s-y)t+k^2+2y^2+2s^2-2yp-4ys+4ps$$

$$Y_1=t^2+2(p+s-y)t+k^2+2y^2-2yp-2ys+2ps$$

$$Y_2=2(p+s-y)k$$

$$Y_3=2(p+s-y)(t+p+s-y)$$

$$Y_4=t^2+2(p+s-y)t+k^2+2ps$$

$k,y,t,p,s$ - integers asked us.

$$a=Y_1^2+Y_2^2$$

$$b=Y_3^2+Y_4^2$$