If variables appear only linearly, then it is possible to use ExtendedGCD (see page 944) to find all solutions to any system of Diophantine equations—or to show that none exist. Particularly from the work of Carl Friedrich Gauss around 1800 there emerged a procedure to find solutions to any quadratic Diophantine equation in two variables—in effect by reduction to the Pell equation x2==a y2+1 (see page 944), and then computing ContinuedFraction[Sqrt[a]]. The minimal solutions can be large; the largest ones for successive coefficient sizes are given below. (With size s coefficients it is for example known that the solutions must always be less than (14 s)5s.).
There is a fairly complete theory of homogeneous quadratic Diophantine equations with three variables, and on the basis of results from the early and mid-1900s a finite procedure should in principle be able to handle quadratic Diophantine equations with any number of variables. (The same is not true of simultaneous quadratic Diophantine equations, and indeed with a vector x of just a few variables, a system m . x2 == a of such equations could quite possibly show undecidability.)
Ever since antiquity there have been an increasing number of scattered results about Diophantine equations involving higher powers. In 1909 Axel Thue showed that any equation of the form p[x,y] == a, where p[x, y] is a homogeneous irreducible polynomial of degree at least 3 (such as x3+x y2 + y3) can have only a finite number of integer solutions. (He did this by formally factoring p[x, y] into terms x-α,i y, then looking at rational approximations to the algebraic numbers α,i.) In 1966 Alan Baker then proved an explicit upper bound on such solutions, thereby establishing that in principle they can be found by a finite search procedure. (The proof is based on having bounds for how close to zero Sum[αiLog[αj], i, j] can be for independent algebraic numbers αk.) His bound was roughly Exp[(c s)105)]—but later work in essence reduced this, and by the 1990s practical algorithms were being developed. (Even with a bound of 10100, rational approximations to real number results can quickly give the candidates that need to be tested.)
Starting in the late 1800s and continuing ever since a series of progressively more sophisticated geometric and algebraic views of Diophantine equations have developed. These have led for example to the 1993 proof of Fermat's Last Theorem and to the 1983 Faltings theorem (Mordell conjecture) that the topology of the algebraic surface formed by allowing variables to take on complex values determines whether a Diophantine equation has only a finite number of rational solutions—and shows for example that this is the case for any equation of the form xn==a yn + 1 with n>3. Extensive work has been done since the early 1900s on so-called elliptic curve equations such as x2 == a y3 + b whose corresponding algebraic surface has a single hole (genus 1). (A crucial feature is that given any two rational solutions to such equations, a third can always be found by a simple geometrical construction.) By the 1990s explicit algorithms for such equations were being developed—with bounds on solutions being found by Baker's method (see above). In the late 1990s similar methods were applied to superelliptic (e.g. xn==p[y]) and hyperelliptic (e.g. x2==p[y]) equations involving higher powers, and it now at least definitely seems possible to handle any two-variable cubic Diophantine equation with a finite procedure. Knowing whether Baker's method can be made to work for any particular class of equations involves, however, seeing whether certain rather elaborate algebraic constructions can be done—and this may perhaps in general be undecidable. Most likely there are already equations of degree 4 where Baker's method cannot be used—perhaps ones like x3==y4 + x y + a. But in recent years there have begun to be results by other methods about two-variable Diophantine equations, giving, for example, general upper bounds on the number of possible solutions. And although this has now led to the assumption that all two-variable Diophantine equations will eventually be resolved, based on the results of this book I would not be surprised if in fact undecidability and universality appeared in such equations—even perhaps at degree 4 with fairly small coefficients.
The vast majority of work on Diophantine equations has been for the case of two variables (or three for some homogeneous equations). No clear analog of Baker's method is known beyond two variables, and my suspicion is that with three variables undecidability and universality may already be present even in cubic equations.
As mentioned in the main text, proving that even simple specific Diophantine equations have no solutions can be very difficult. Obvious methods involve for example showing that no solutions exist for real variables, or for variables reduced modulo some n. (For quadratic equations Hasse's Principle implies that if no solutions exist for any n then there are no solutions for ordinary integers—but a cubic like 3x3 + 4 y3 + 5z3 == 0 is a counterexample.) If one can find a bound on solutions—say by Baker's method—then one can also try to show that no values below this bound are actually solutions. Over the history of number theory the sophistication of equations for which proofs of no solutions can be given has gradually increased—though even now it is state of the art to show say that x==y==1 is the only solution to x2 == 3 y4 -2.
Just as for all sorts of other systems with complex behavior, some idea of overall properties of Diophantine equations can be found on the basis of an approximation of perfect randomness. Writing equations in the form p[x1, x2, …, xn]==0 the distribution of values of p will in general be complicated (see page 1161), but as a first approximation one can try taking it to be purely random. (Versions of this for large numbers of variables are validated by the so-called circle method from the early 1900s.) If p has total degree d then with xi < x the values of Abs[p] will range up to about xd. But with n variables the number of different cases sampled for xi < x will be xn. The assumption of perfect randomness then suggests that for d<n, more and more cases with p==0 will be seen as x increases, so that the equation will have an infinite number of solutions. For d>n, on the other hand, it suggests that there will often be no solutions, and that any solutions that exist will usually be small. In the boundary case d==n it suggests that even for arbitrarily large x an average of about one solution should exist—suggesting that the smallest solution may be very large, and presumably explaining the presence of so many large solutions in the n=d=2 and n=d=3 examples in the main text. Note that even though large solutions may be rare when d>n they must always exist in at least some cases whenever there is undecidability and universality in a class of equations. (See also page 1161.)
If one wants to enumerate all possible Diophantine equations there are many ways to do this, assigning different weights to numbers of variables, and sizes of coefficients and of exponents. But with several ways I have tried, it seems that of the first few million equations, the vast majority have no solutions—and this can in most cases be established by fairly elementary methods that are presumably within Peano arithmetic. When solutions do exist, most are fairly small. But as one continues the enumeration there are increasingly a few equations that seem more and more difficult to handle.