There have been numerous papers over the last few years that have
been concerned with sequences of integers for which `Kummer-type' Theorems,
or `Lucas-type' Theorems or `Wolstenholme-type' Theorems hold.
One nice example is the Apéry numbers,
which were introduced in Apéry's proof of the irrationality of .
At first, a few seemingly surprising congruences were found for these numbers,
but in 1982, Gessel  showed that these were
all consequences of the fact that the Apéry numbers satisfy
`Lucas--type' and `Wolstenholme--type' Theorems
(that is and for all and primes ).
R. McIntosh has asked whether a non--trivial sequence of integers,
satisfying a `Lucas--type' Theorem, can grow slower than ?