Are Primes Really Random?

Number Theory Level 3

In general, about $\frac{1}{4}$ of all primes less than $n$ end in a 1, 3, 7, and 9, respectively. Would the same symmetry hold if we considered the last digits in pairs of consecutive primes less than, say, $n=100000?$

Yes, definitely Probably not

1 solution

Eli Ross Staff
Mar 25, 2016

Surprisingly, this is probably not the case, according to recent observations by Robert Lemke Oliver and Kannan Soundarajan. In particular, they found that primes "really hate to repeat themselves"; for example, a prime followed by a 1 is more likely to be followed by a prime ending in a 3, 7, or 9 rather than another prime ending with a 1.

A quick explanation of this is, "well, we get to check 3 numbers ending in 3,7,9 for primality before we get to see another number ending in a 1." However, this actually wouldn't explain the magnitude of the bias they found in their data. Instead, there may be deeper connections between Lemke Oliver and Soundarajan's observation and the Hardy-Littlewood prime $k$ -tuple conjectures and it will likely take more research to figure out exactly what's going on here!

The symmetry still exists for consecutive primes! Just look at Conjecture 1.1: for large $x$ , all the bias terms decay to zero and the asymptotic behaviour of $\pi(x;q,\mathbf a)$ is still dominated by the $\text{li}(x)/\phi(q)^r$ term predicted by symmetry.

The point of the paper is that the bias terms take much, much longer to decay to zero than one expects from the observed bias in the single-prime case. But for very large $n$ , we do expect to see $\tfrac1{16}$ th of each possible pair of ending digits (conjecturally, as we have very few theorems for this).

Erick Wong - 5 years, 2 months ago

Thanks Erick - I think the question is clearer now. For anyone interested, this is the paper .

Eli Ross Staff - 5 years, 2 months ago

Are Primes Really Random?

1 solution

0 pending reports