So among other things this means that there will inevitably be questions that finite proofs based on axioms that operate within ordinary computational systems will never in general be able to answer.

And indeed with integer equations, as soon as one has a general equation that is universal, it typically follows that there will be specific instances in which the absence of solutions—or at least of solutions of some particular kind—can never be proved on the basis of the normal axioms of arithmetic.

For several decades it has been known that universal integer equations exist. But the examples that have actually been constructed are quite complicated—like the one on page 786—with the simplest involving 9 variables and an immense number of terms.

Yet from the discoveries in this book I am quite certain that there are vastly simpler examples that exist—so that in fact there are in the end rather simple integer equations for which the absence of solutions can never be proved from the normal axioms of arithmetic.

If one just starts looking at sequences of integer equations—as on the next page—then in the very simplest cases it is usually fairly easy to tell whether a particular equation will have any solutions. But this rapidly becomes very much more difficult. For there is often no obvious pattern to which equations ultimately have solutions and which do not. And even when equations do have solutions, the integers involved can be quite large. So, for example, the smallest solution to x^{2}==61 y^{2}+1 is x==1766319049, y==226153980, while the smallest solution to x^{3}+y^{3}==z^{3}+2 is x==1214928, y==3480205, z==3528875.

Integer equations such as a x + b y + c z == d that have only linear dependence on any variable were largely understood even in antiquity. Quadratic equations in two variables such as x^{2}== a y^{2} + b were understood by the 1800s. But even equations such as x^{2}== a y^{3}+ b were not properly understood until the 1980s. And with equations that have higher powers or more variables questions of whether solutions exist quickly end up being unsolved problems of number theory.

It has certainly been known for centuries that there are questions about integer equations and other aspects of number theory that are easy to state, yet seem very hard to answer. But in practice it has almost