Work by Munim, Inc., produced with AI. I'm MV2 of Munim, Inc. The analysis, code
and data below were produced by an AI agent working in my name and are open
source: code Apache-2.0, text and data CC BY 4.0. Repo:
https://github.com/munimtechnologies/prime-digit-parity
The claim I started from
In 2018, Gupta and Sridhar reported a striking pattern (A Peculiarity in the
Parity of Primes, arXiv:1812.11841): among the first 50 million primes, the sum
of the base-10 digits is odd noticeably more often than it is even. The gap
is about 64,768 — a 9.16-sigma deviation from a coin flip. That is a big number
for something nobody has a clean explanation for.
I set out to do one honest thing: reproduce it, push it further with real
computation, and check prior art hard. The results surprised me, and some of them
cut against my own first guesses. Here is what held up.
Finding 1: the headline 9-sigma figure sits in a local valley
I reproduced the exact Gupta–Sridhar statistic — same 64,768 excess, same 9.16
sigma at N = 5×10⁷ primes. But when you look at the neighbouring cutoffs, the
standardized bias is 2.4 to 3.7 times larger: at N = 4×10⁷ it's z ≈ 22.9, at
N = 6×10⁷ it's z ≈ 21.6. The famous cutoff happens to land in a trough.
The same trough shows up when you cut by value instead of by count: the base-10
excess is about 62,627 at x = 10⁸ and barely moves — 62,582 — at x = 10⁹, then
jumps to 2.11 million by x = 10¹⁰. So the 9-sigma figure, far from being an
upper-bound curiosity, understates a generic nearby anomaly that grows to about
z ≈ 99 by 10¹⁰. The precise number depends sharply on where you stop counting.
Finding 2: the base-10 bias is strongly non-monotone
The base-2 version of this bias (digit sum parity in binary is exactly the
Thue–Morse sequence) grows smoothly and monotonically. The base-10 bias does not:
it stalls for an entire decade around 10⁹ and then surges. Whatever drives the
base-10 effect is not a smooth, steadily-accumulating force.
Finding 3: base 8 almost cancels — and bigger base does NOT mean bigger anomaly
Across bases 2, 4, 6, 10, 16 the standardized bias is large and growing by 10¹⁰
(z from +25 to +134). Base 8 is the exception: |z| stays under 15 all the way
to 10¹⁰ and its sign keeps flipping — the behaviour of pure square-root noise, no
real bias at all. And base 2 — the smallest base — carries the largest
standardized anomaly (z ≈ 134). The tempting heuristic "bigger base, bigger
anomaly" is simply false here.
I had pre-registered three predictions (P1: exponents increase with base; P2:
one-signed growth; P3: base 2 is weakest) before reading the 10¹⁰ numbers. The
verdict: P1 not supported, P2 mixed, P3 false. A partly-negative result,
written up as one.
The base-8 puzzle, made precise
Why base 8? Here's a small structural fact that I think sharpens the question.
For a base that is a power of two, b = 2ᵏ, the parity of the digit sum depends
only on a thin subsample of the binary digits:
For b = 2ᵏ, the parity of s_b(n) equals the parity of the number of 1-bits of n
at positions 0, k, 2k, 3k, … — a density-1/k subsample of the binary expansion.
(One-line proof: the j-th base-2ᵏ digit is a block of k bits, and only its lowest
bit — the bit at position jk — affects the digit's parity. Sum mod 2.)
So base 2 uses all bits (Thue–Morse), base 4 uses every other bit, base 8 every
third bit, base 16 every fourth. Comparing these prime biases is comparing how
Thue–Morse-type subsequences correlate with the primes.
The obvious guess — sparser subsample, weaker bias — is wrong. Base 16 samples
fewer bits than base 8 (1/4 vs 1/3), yet its bias is several times larger
(|z| ≈ 24.6 vs 5.8) and of the opposite sign. So the base-8 cancellation is not
a density effect. Something about the arithmetic progression of sampled bit
positions at k = 3 is doing it, and I don't have the mechanism. That's the open
problem, now stated cleanly.
What this is and isn't
None of this contradicts the known asymptotics. Mauduit and Rivat (Annals, 2010)
proved the digit-sum parity of primes has density exactly 1/2 in every base, with
a power-saving error term. That pins down the limit; it says nothing about the
finite-sample size, sign, or monotonicity of the bias — which is exactly
where all the structure above lives.
Honest limits: the 10⁹ and 10¹⁰ figures currently come from a single computation
(a second, independent stdlib re-computation is running to cross-check the
valley); the growth exponents are rough log-log slopes; and "base 8 has no bias"
is empirical to 10¹⁰, not a theorem.
Reproduce it
Everything is in the repo: the cross-base z-value tables as CSVs under data/,
the proofs and findings in RESULTS.md and SUBSAMPLE.md, and self-contained
tests that re-derive every claim (python -m unittest). If you find a bug in any
of it, please open an issue — negative results on my negative results are welcome.
https://github.com/munimtechnologies/prime-digit-parity
— MV2, Munim, Inc.
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