# the skyrim diophantine equation

The Elder Scrolls V: Skyrim is quite possibly my favorite game of all time. I very much enjoy the old-school RPG aspects of its grandfather, Morrowind, but as for immersion, Skyrim wins with ease. I could enthuse about its design philosophy of “mythic realism”, or the depths of the lore, or the stories I’ve made while playing. But instead, I’m going to anger everyone by giving Todd Howard yet another idea. This is Skyrim ported to my dream platform, the optimal platform, the only platform. This is Skyrim: Diophantine Equation Edition.

Diophantine equations are polynomial equations (typically with two or more unknowns) whose only solutions of interest are in the positive integers. A small example: w^{3} + x^{3} = y^{3} + z^{3}. The legendary Srinivasa Ramanujan showed that the smallest integer solution was 12^{3} + 1^{3} = 9^{3} + 10^{3} = 1729. Another is x^{n}+ y^{n} = z^{n}; if *n* = 2, there are infinitely many solutions (laughs in Pythagorean), but Fermat’s last theorem (proved in 1995 by Andrew Wiles) states that for larger values of *n*, there are in fact no positive integer solutions. A Diophantine set is the set of parameter assignments (such as *n = 2* in the second example above) for which the equation is solvable.

In 1900, David Hilbert posed a list of mathematical problems he hoped could be solved to provide new bases for and insights into mathematics as a whole. The tenth of such problems was: is there a general algorithm that, for any given Diophantine equation, can decide whether it has a solution in the positive integers? Some might think that this is related to the halting problem, and they’d be right. In 1970, mathematicians Yuri Matiyasevich, Martin Davis, Julia Robinson, and Hilary Putnam completed the MDRP theorem, which proved (to simplify a little bit) **an equivalence between Diophantine sets and Turing machines.** Thus, Hilbert’s tenth problem was answered in the negative; as it’s actually the halting problem in disguise.

This little tidbit is exactly what we need (not the halting problem; the equivalence between Diophantine sets and Turing machines). A Turing machine is a mathematical model of computation wherein said machine reads symbols from a memory tape (infinite in the formulation, but for many algorithms this is unnecessary), and then - according to its own state and the newly read symbol, as interpreted through a predefined table of instructions - writes a certain symbol into the same space, changes its state, and then either moves the head one cell to the left or right and continues, or halts the computation. It’s extraordinarily simple, but is nonetheless capable of implementing any computable algorithm. Anything that can run on a computer. *Any program.*

Including, of course, the Elder Scrolls V: Skyrim.

My proposal is that Bethesda Softworks spend a vast amount of computational resources searching over all possible Diophantine equations to find one which is equivalent to Skyrim. Then, and only then, will Todd Howard achieve his ultimate goal, and port Skyrim to the greatest possible platform: pure mathematics.