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  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  who established the existence of another solution. The equation attracted interest of other researches, Daniel Shanks  was first in writing down two solutions explicitly and later he and Samuel S. Wagstaff, Jr.  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.