# on sums of squares being squares

A sum of 2 squares can be a square, eg. $3^2+4^2=5^2$.

A sum of 3 squares can be a square, eg. $3^2+4^2+12^2=13^2$, but is it possible to find an example where sum of any two of them would also be a square?

What about 4 squares such that sum of any two, any three and all four are squares?

Are there arbitrarily long sequences of squares such that any sum among them is a square?

• What are your thoughts/attempts? – Jack D'Aurizio Jul 13 '17 at 19:06
• I checked with computer, that there are no such triples with all numbers less than 150. It's my own problem stated while studying Pythagorean triples, I'm not able to solve it, I'm interested in what is known about it – tong_nor Jul 13 '17 at 19:34
• Why ask such questions? You don't understand how to solve a system of nonlinear Diophantine equations. Ask a question - not realizing how it is difficult. And you want to get a simple answer. – individ Jul 14 '17 at 4:31
• Your second sentence in symbols would be $$a^2+b^2+c^2=d^2\\ a^2+b^2 = \square_1\\ a^2+c^2 = \square_2\\ b^2+c^2 = \square_3$$ This is the perfect cuboid problem and, as pointed out by Somos, is still an open problem. – Tito Piezas III Jul 10 at 6:51