# Definition talk:P-adic Valuation/Integers

Regarding the suggestion of a merge Definition:Prime Decomposition/Multiplicity it should be noted that the multiplicity of $p$ is only defined if $p$ is a prime factor of an integer, where as the $p$-adic valuation is always defined for every integer. The multiplicity of $p$ is never 0, but the $p$-adic valuation can be. This subtle difference means that the two definitions are not the same. --Leigh.Samphier (talk) 12:52, 7 April 2021 (UTC)
You could create a theorem that shows that for any integer the $p$-adic valuation equals the multiplicity if $p$ divides the integer and 0 otherwise. This states explicitly the relationship between the two definitions. The page could be used to emphasise that the multiplicity for $p$ is not defined if $p$ does not divide an integer. --Leigh.Samphier (talk) 10:09, 8 April 2021 (UTC)