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A New Approach to the First Digit Phenomena

Dennis P. Allen
Year: 1999

In this paper, we first show by extending the proof of B. J. Flehinger that the integers have the first digit property that the primes represented to the base ten also have the first digit property. We note that R. E. Whitney has also proven this using the logarithmic matrix method of summability. We then abstract from these two proofs in view of the Peano Axioms to obtain a (new) definition of what it means to sum a sequence in the spirit of the Peano axioms for the positive integers (which includes Flehinger's and Whitney's methods as special cases) and conjecture any method of summation of this very general type assigns the limit log10((A+ 1)/A) to the two sequences sn and tn where

  1. sn = 1 if n has first digit equal to A else 0, and
  2. tn = 1 if the nth prime has first digit A else 0.

The conjecture, if true, yields a new explanation of the first digit phenomenon by reducing it to its first cause: the basic well ordering of the positive integers.