# Using Diophantus' method, find four square numbers such that their sum added to the sum of their sides is 73. Why does Diophantus' method always work?This is on the early usage of algebra, and I...

Using Diophantus' method, find four square numbers such that their sum added to the sum of their sides is 73. Why does Diophantus' method always work?

This is on the early usage of algebra, and I have never learned about Diophantus, what his method is, or how to do it.

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### 1 Answer

This question has been up for awhile, so I'll just give you what I have on it. I couldn't find anything on "Diophantus' Method" online - it sounded like he used lots of different ways to solve equations.

He was only interested in positive rational solutions, but I found a proof that there are no positive integer solutions, so I'll give you that.

We're interested in `a^2+b^2+c^2+d^2+a+b+c+d=73` (I think that's what you're saying).

`=a(a+1)+b(b+1)+c(c+1)+d(d+1)=73`

Now, if `a` is even, then `a+1` is odd, and vice versa. The product of an even and an odd is always even, so `a(a+1)` is always even. The same goes for the rest of the terms, so we have

even + even + even + even = 73

This told me to stop trying to guess and check!

Anyway, let us know what your instructor says the correct answer is.