Some Problems from Diophantus’s Arithmetika

http://www.math.uconn.edu/~leibowitz/math2720f08/diophantusexamples.pdf

Diophantus of Alexandria

http://www.robertnowlan.com/pdfs/Diophantus%20of%20Alexandria.pdf

One of his propositions states that the difference of the cubes of two numbers can always be expressed as the sum of the cubes of two other numbers; that no number of the form 4n – 1 can be expressed as the sum of two squares; and that no number of the form 8n – 1 can be expressed as the sum of three squares


Given two cubes, to find in rational numbers two others such that their difference is equal to the sum of the given cubes.

That is to say, solving a^3 + b^3 = x^2 - y^2

x = a(a^3 + 2b^3) / (a^3 - b^3)
y = b(2a^3 + b^3) / (a^3 - b^3)

Given two cubes, to find in rational numbers two cubes such that their difference is equal to the difference of the given cubes.


That is to say, a^3 - b^3 = x^3 - y^3

Vieta finds

x = b(2a^3 - b^3) / (a^3 + b^3)
y = a(2b^3 - a^3) / (a^3 + b^3)