Diophantine Equations

Today I turned 24521 days old and it turns out that this number is a non-negative value of x in the solution (x, y) to the Diophantine equation: 

x^2+(x+16807)^2=y^2

The corresponding y value is 48055 so the solution is (24521, 48055). So I thought that this would be an appropriate time to include some information about Diophantine equations. According to Wikipedia:

The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis.

 The definition given in the same article is:

In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer solution is a solution such that all the unknowns take integer values). A linear Diophantine equation is an equation between two sums of monomials of degree zero or one. An exponential Diophantine equation is one in which exponents on terms can be unknowns.

Today's Diophantine equation is quadratic and it's equivalent to finding integer solutions to the right-angled triangle with hypotenuse y and arms x and x + 16807. As it turns out, the requisite triangle is 48055, 24521 and 41328. In general terms, finding all right triangles with integer side-lengths is equivalent to solving the Diophantine equation a^2 + b^2 = c^2, except in this case we know that b = a + 16807.

Diophantine analysis is a big topic to which I may return at a later date but that's enough for now.