Silent Primes

Some primes produce zero autocorrelation at a given lag. No digit matches its shifted counterpart. The repetend and its rotation are completely unrelated. These primes are silent at that lag.

Which primes are silent?

The gate width theorem says $b - 1$ lags are deranging at every prime. But different primes are deranging at different specific lags. A prime that is silent at lag $\ell$ has $b^\ell \bmod p$ in the deranging set.

Fix a lag $\ell$. The deranging condition is: $c = b(1 - b^\ell){-1} \bmod p$ falls in ${1, \ldots, b-1}$. For each $c$ in that range, the primes $p$ satisfying this condition form an arithmetic sequence. The silent primes at a given lag are the prime members of finitely many explicit arithmetic progressions.

There are only finitely many silent primes at each lag (for fixed base and lag, eventually every prime is non-silent). The digit function becomes noisier as $p$ grows: more positions match, fewer lags are completely silent.

The variance

Among the non-deranging multipliers (the ones with $C(g) > 0$), how much does the collision count vary? The answer: the standard deviation scales as $\sqrt{Q}$, where $Q = \lfloor(p-1)/b\rfloor$ is the bin size.

The square-root scaling is the signature of equidistribution on the finite field. The collision count fluctuates like a random variable whose range is the bin size. The deviation from the mean follows the same law that governs character sums, lattice point counts, and other quantities determined by equidistribution.

The Dirichlet series

Define $L^G(s) = \sum_p C(g_p) p^{-s}$ where $g_p$ is a fixed multiplier evaluated at each prime. This series encodes the collision count across the population of primes. Its convergence behavior at $s = 1$ is the collision analogue of the prime number theorem.

The silent primes are where $C(g_p) = 0$. They contribute nothing to the Dirichlet series. The non-silent primes carry the signal.

Try it yourself

./nfield 53              # p=53: silent at some lags
./nfield 7               # p=7: DP prime, silent at all non-trivial lags

Code: github.com/alexspetty/nfield

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Alexander S. Petty
August 2024
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