ABSTRACT
IfF(i') = 1 for too many i < h’s then n may not be logarithmic and we cannot apply Theorem 7.2 in order to define the p -adic valuation of n,-</i ^ (0 - We give the following equivalent definition of /?-adic valuation of y = n,-<#i ^ ( 0 -
where \A\ denotes the cardinality of a set A. In other words, we count the number of is less than n for which F(i) > 1. In this way we reduce the computation of G{n) = ]!/<« ^ ( 0 to the case when F is always > 1 and therefore G grows exponentially in n. This approach needs to be handled with some care since in /Ao we cannot count; there is not a good notion of cardinality. This is again related to the problem of the nontotality of exponentiation: in order to formalize the notion of cardinality we need to use functions and we may not be able to code them. The following is still an open problem.
