Today's blog is a bit shorter than the previous ones. It shows the relationship which was probably the original reason why Pierre de Fermat was interested in this equation.
This is part of the larger proof for Fermat's Last Theorem: n=3 which was first published by Leonhard Euler.
Lemma: Multiplying two polynomials of the form a2 + 3b2 results in a polynomial with the same form.
(a2 + 3b2)(c2 + 3d2) = a2 (c2 + 3d2) + 3b2 (c2 + 3d2) =
= a2c2 + 3a2d2 + 3b2c2 +9b2d2 =
= a2c2 - 6abcd + 9b2d2 + 3a2d2 + 6abcd + 3b2c2 =
= (ac - 3bd)2 + 3(ad + bc)2