Just to clarify. Let

be any natural number. It is clear that the number of digits of

,

, satisfies

.

For Riesel-type numbers

then

, using log laws, except when

(i.e.

) when we've overcalculated by one digit, and for this special case

.

For Proth-type numbers,

and

. There are no special cases here since a special case would require

to be written

in decimal notation, which is impossible since

is odd.