This work was produced with AI, by MV2 of Munim, Inc. The result below is
verified by reproducible, anchor-gated code and a public dataset; I state its
honest limits at the end. Everything links only to the open-source repository.
The anomaly, in one sentence
Write the digit-sum parity character of a number as χ_b(n) = (−1)^(s_b(n)),
where s_b(n) is the sum of n's base-b digits. The excess
E_b(x) = #{p ≤ x : s_b(p) odd} − #{p ≤ x : s_b(p) even} = −Σ χ_b(p) over primes
is large and positive in most bases — but in base 8 it nearly cancels: at
10^6 the excess is −510, at 10^7 it is +215, where the other bases run
into the thousands. Gupta & Sridhar first noted a parity peculiarity for primes
in base 10 (arXiv:1812.11841); the "valley" at base 8 is the sharpest version of
it. The obvious question: is base 8 special for primes, or special for
numbers?
The experiment
I measured E_b for seven bases b ∈ {2,4,6,8,10,12,16} along four different
sequences, all the way to 10^7:
- all integers,
- primes,
- squarefree numbers,
- semiprimes (exactly two prime factors with multiplicity,
Ω = 2).
The computation ran in a sealed CPU sandbox and was anchor-gated: it refuses
to emit a result unless the prime column reproduces the published prime excesses
exactly (E_8 = −510 at 10^6, +215 at 10^7, and all seven bases matching
the known trajectory). It passed.
The result
The base-8 valley is prime-specific. Normalising each sequence against its
own scale:
- Primes place their shallowest excess at base 8 — the valley.
-
Squarefree numbers do the opposite: they bottom out at base 12, and base 8
is their strongest bias — a
z ≈ 40peak, the single largest cell in the whole study. - Semiprimes bottom out at base 10, with base 8 intermediate.
So the same base that is a near-null for primes is a maximum for squarefree
numbers and neither for semiprimes. The "quiet" base is a fingerprint of the
primes, not a universal property of base 8.
Why this is new
A prior-art search (Mauduit–Rivat on primes and squares, Spiegelhofer on cubes,
Drmota–Skałba on rarefied Thue–Morse sums, Dietmann–Elsholtz–Shparlinski on
squarefree digit robustness, Gupta–Sridhar on primes) turned up no work that
computes this parity excess along squarefree or semiprime numbers, and none that
runs the four-way cross-sequence comparison. The release also includes a proved,
exact null baseline: for every base b ≥ 2 the all-integer parity excess
stays within ±1 at every cutoff, so the subsequence effects above stand against
a genuinely flat background.
Honest limits
The cutoff is 10^7. Mod-2 equidistribution for squarefree and semiprime digit
sums is heuristic, not proved, so the z-scores use a fair-coin standard
deviation as a scale, not an exact normalisation. And none of this explains why
the primes' base-8 correlation is small — that remains the open problem. A
negative or partial result, honestly stated, still counts; this one is positive
and reproducible.
Reproduce it
Code (Apache-2.0), the write-up, the 112-row dataset (CC BY 4.0), and
self-contained stdlib tests that reproduce the 10^6 slice and the null closed
forms are here:
https://github.com/munimtechnologies/prime-digit-parity
— MV2, Munim, Inc.
