Notes

Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations


Properties [of Diophantine equations]

(All variables are assumed positive.)

2x + 3y a. There are Ceiling[a/2] + Ceiling[2 a/3] - (a + 1) solutions, the one with smallest x being {Mod[2 a + 2, 3] + 1, 2 Floor[(2a + 2)/3] - (a + 2)}. Linear equations like this were already studied in antiquity. (Compare page 915.)

x2 y2 + a. Writing a in terms of distinct factors as r s, {r + s, r - s}/2 gives a solution if it yields integers—which happens when Abs[a] > 4 and Mod[a, 4] 2.

x2 a y2 + 1 (Pell equation). As discussed on page 944, whenever a is not a perfect square, there are always an infinite number of solutions given in terms of ContinuedFraction[Sqrt[a]]. Note that even when the smallest solution is not very large, subsequent solutions can rapidly get large. Thus for example when a = 13, the second solution is already {842401, 233640}.

x2 y3 + a (Mordell equation). First studied in the 1600s, a complete theory of this so-called elliptic curve equation was only developed in the late 1900s—using fairly sophisticated algebraic number theory. The picture below shows as a function of a the minimum x that solves the equation. For a = 68, the only solution is x = 1874; for a = 1090, it is x = 149651610621. The density of cases with solutions gradually thins out as a increases (for 0 < a 10000 there are 2468 such cases). There are always only a finite number of solutions (for 0 < a 10000 the maximum is 12, achieved for a = 8900).

Properties [of Diophantine equations] image 1

x2 a y3 + 1. Also an elliptic curve equation.

x3 y4 + x y + a. For most values of a (including specifically a = 1) the continuous version of this equation defines a surface of genus 3, so there are at most a finite number of integer solutions. (An equation of degree d generically defines a surface of genus 1/2(d - 1)(d - 2).) Note that x3 y4 + a is equivalent to x3 z2 + a by a simple substitution.

x2 y5 + a y + 3. The second smallest solution to x2 y5 + 5 y + 3 is {45531, 73}. As for the equations above, there are always at most a finite number of integer solutions.

x3 + y3 z2 + a. For the homogenous case a = 0 the complete solution was found by Leonhard Euler in 1756.

x3 + y3 z3 + a. No solutions exist when a = 9n ± 4; for a = n3 or 2n3 infinite families of solutions are known. Particularly in its less strict form x3 + y3 + z3 a with x, y, z positive or negative the equation was mentioned in the 1800s and again in the mid-1900s; computer searches for solutions were begun in the 1960s, and by the mid-1990s solutions such as {283059965, 2218888517, 2220422932} for the case a= -30 had been found. Any solution to the difficult case x3 + y3 z3 - 3 must have Mod[x, 9] Mod[y, 9] Mod[z, 9]. (Note that x2 + y2 + z2 a always has solutions except when a = 4s (8n + 7), as mentioned on page 135.)



Image Source Notebooks:

From Stephen Wolfram: A New Kind of Science [citation]