Lemke Oliver's Dodgy Primes
On Friday, Robert J. Lemke Oliver presented a talk at the UVM Mathematics Colloquium entitled “Prime Numbers, Randomness, and the Gambler’s Fallacy.”
Abstract: “Prime numbers are often said to be ‘random’, but, given that primes are deterministic, what does that actually mean? One way in which this randomness manifests is in the last digits of primes: it turns out that each possible last digit is equally likely in a certain strong sense. A similar story holds for the residue class of primes modulo any fixed integer, and this is a well-understood classical theorem of analytic number theory. Surprisingly, however, in joint work with K. Soundararajan, we find that an analogous phenomenon does not hold for patterns of consecutive primes. For example, a string of consecutive primes ending in the digit 1 strongly predisposes the following prime to not end in a 1; thus, prime numbers are subject to the gambler’s fallacy. This talk will be aimed at the level of graduate students and non-experts, but should be satisfying to practicing number theorists as well.”
What Lemke Oliver presented was, in a word, peculiar. While the last digits of primes are randomly distributed in the limit, there is considerable bias that adjacent primes tend not to be in the same residue class at smaller values. Moreover, this bias decreases very, very slowly (like $$\frac{log ; log ; x}{log ; x}$$ which ain’t going anywhere any time soon). Weird, huh?
Related reading: From Prime Numbers to Nuclear Physics and Beyond, Institute for Advanced Study, Princeton University