Single-Fold Representations

as a consequence each known Diophantine representation of exponentiation is infinite-fold—as soon as the corresponding equation (2.9) has a solution, it has infinitely many of them.

Single-fold representations have important applications (one of them is given in Sect.2.6.2), and for this reason Martin Davis paper [7] titled “One equation to rule them all” remains of interest. The equation from the title is

and it has a trivial solution u = r = 1, v = s = 0. Martin Davis proved that if this is the only solution, then some Diophantine relation has exponential growth. His expectations were broken by Oskar Herrman [17] who established the existence of another solution. The equation attracted interest of other researches, Daniel Shanks [53] was first in writing down two solutions explicitly and later he and Samuel S. Wagstaff, Jr. [54] found 48 more solutions.

The discovery of non-trivial solutions did not spoil Martin Davis approach completely. It fact, it can be shown that if (2.15) has only finitely many solutions then every listable set has a single-fold Diophantine representation.

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