A graph illustrating the “eight-digit gap”

This illustrates the base-10 logarithm (y-axis) of the nth century with fifteen, sixteen or seventeen primes up to n=365. Note the steep logarithmic rises within and around the “eight-digit gap”

In several previous posts I have discussed a phenomenon that I have come to call the “eight-digit gap”, by which the maximum observed count of prime numbers per century decreases up to and including centuries with eight digits and then increases. Although the proportion of primes is consistently decreasing, for numbers with more than eight digits the decrease in proportion of primes is slower than the rise in the number of centuries. Consequently, even with a smaller proportion of primes there are so many more centuries with large numbers of total primes for digits counts far above the “eight-digit gap”.

The “eight-digit gap” begins approximately at three million [of course those numbers have seven digits, but the minimum begins after the prime-rich sequences starting at 2,704,900 and 2,967,310]. As early as the pioneering studies of James Glaisher, it was noted that the maximum number of centuries in the fourth million [from 3,000,000 to 3,999,999] was less than in the earlier millions. Glaisher’s studies extended only into the “core” of the “eight-digit gap” beginning at 7,587,800, and not far into that “core” area [which extends up to 23,742,099 and contains no century with more than fourteen primes].

The graph shown, though a little difficult to read because it is extended to so many centuries with magnitudes far larger than numbers lying within the “eight-digit gap”, clearly shows the highly vertical lines of the logarithmic plot in the region, and the much flatter plots above it and to a much smaller extent below it. The vertical plots for 16 and 17 primes illustrate the “jump” across the “eight-digit gap” extremely well.