REVIEW 3 minor 4 references
On the $\mathcal{A}$-transcendence of a Champernowne-type constant
T0 review · 0 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash
Pith's one-line read For every nonzero integer polynomial f, infinitely many n have p_n not dividing f(n), settling the strong-transcendence conjecture for the Champernowne-type element π(p).
desk verdict Correct proof of the Matsusaka–Seki conjecture, with a surprising Maynard–Tao input; minor presentation issues but no mathematical gaps. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The argument rests on three tools. The first is the Maynard–Tao bounded-gaps theorem: for any fixed k, liminf_{n→∞}(p_{n+k} − p_n) < ∞, which supplies an infinite set M on which k+1 consecutive primes have a fixed offset pattern. The second is a linear-system construction: the coefficients c_j are chosen to satisfy Σ_j c_j j^u h_j^v = 0 for 0≤u≤d, 0≤v≤d−1, making the main terms in a geometric-series expansion cancel. The third is the two-variable polynomial Φ(x,y) = Σ_j c_j f(x+j) ρ(y)/(y+h_j), which evaluates to a rational integer divided by ρ(p_n) on M, and whose leading term dominates as n → ∞.
What would settle it
Exhibit a nonzero f(x) ∈ Z[x] and an integer N such that p_n divides f(n) for every n ≥ N; Theorem 1.1 denies the existence of any such pair. The paper shows this assumption forces a nonzero two-variable polynomial to vanish on an infinite set, so writing down such an f and N would refute the theorem.
Extended reading notes
Core claim
The central claim is Theorem 1.1: for every nonzero f(x) ∈ Z[x], the set {n > 0 : p_n ∤ f(n)} is infinite. The paper shows this by contradiction: assuming p_n | f(n) for all large n, it uses the bounded-gaps theorem to find infinitely many n for which a fixed block of k+1 consecutive primes has fixed offsets h_0,...,h_k. It then constructs a nonzero polynomial Φ(x,y) with integer coefficients such that, on that infinite set, Φ(n,p_n)/ρ(p_n) is an integer that is also o(1), forcing Φ(n,p_n)=0 eventually, which contradicts the leading-term growth of Φ. Since the existence of such f would mean π(p) satisfies a nontrivial polynomial relation over Q, the conclusion is that π(p) ∉ C_A, the integra
Load-bearing premise
The proof depends on the Maynard–Tao bounded-gaps theorem for the specific gap parameter k = d(d+1); if that theorem failed to provide infinitely many blocks of k+1 consecutive primes within a bounded distance, the contradiction argument would collapse.
Editorial extensions
If this is right
- The Champernowne-type constant π(p) = (n mod p_n)_p is not algebraic over Q in the ring A; it is transcendental in the strong sense of lying outside the integral closure C_A.
- Conjecture 6.2 of the earlier paper [2] is settled in the affirmative, completing a line of partial results that only covered low-degree polynomials or conditional assumptions.
- No nonzero integer polynomial f can have the property that p_n divides f(n) for all sufficiently large n; this gives a new, elementary-looking statement about primes and polynomial values.
- The proof shows that the obstruction to algebraic relations is not the growth of the representatives but the combinatorial structure of the prime-indexed values, since the cancellation works for arbitrarily high degree.
Reading between the lines
- The same cancellation argument may extend to prove analogous non-algebraicity statements for other elements of A whose components are defined by prime indices n and an associated polynomial, provided a bounded-gaps supply of offset patterns is available.
- The method suggests a possible route to showing that π(p) is not only outside C_A but also satisfies no algebraic differential equation at the level of the components, though this is not explored in the paper.
- One could test the robustness by replacing the prime sequence with other sequences (e.g., primes in arithmetic progressions) and asking whether the analogue of Champernowne's constant remains strongly transcendental; the bounded-gaps theorem may supply the needed structure there as well.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves Theorem 1.1: for every nonzero polynomial f(x) in Z[x], there are infinitely many positive integers n with p_n ∤ f(n). The proof is a contradiction argument. Assuming p_n | f(n) eventually, it sets k = d(d+1), where d = deg f, and applies the Maynard–Tao theorem to obtain infinitely many n for which p_n, ..., p_{n+k} have a fixed tuple of offsets h_0=0<...<h_k. A nontrivial integer solution c_j of the homogeneous linear system (2.2) is used to define a nonzero two-variable polynomial Φ(x,y). For the infinite set M of such n, the quotient Φ(n,p_n)/ρ(p_n) is shown to be an integer. A finite geometric-series expansion plus the linear relations (2.2) cancels the main terms, leaving O(n^d/p_n^{d+1}) = o(1); hence the integer quotient is 0 for all sufficiently large n in M. This contradicts the nonvanishing asymptotic (2.3) obtained from the prime number theorem. The paper states that this theorem is equivalent to [2, Conjecture 6.2], which asserts the strong transcendence π(p)∉C_A of the Champernowne-type element.
Significance. The result is significant: if correct, it settles Conjecture 6.2 of Matsusaka–Seki and gives a short, transparent proof that the Champernowne-type element π(p) is not algebraic over Q. The proof is fully explicit and has no fitted constants or hidden numerical coincidences; it relies only on the prime number theorem and the Maynard–Tao theorem, both cited precisely. The reduction to a finite homogeneous linear system is elegant, and the cancellation argument is complete. The paper substantially improves the partial results in [2] (degree at most two, or conditional cases). The heavy use of Maynard–Tao is legitimate because the theorem is exactly what is needed to force a fixed finite tuple of gaps infinitely often. I regard the mathematical content as sound and the presentation as sufficiently clear for publication.
minor comments (3)
- [Introduction, last paragraph] The statement that Conjecture 6.2 of [2] is equivalent to Theorem 1.1 is asserted without proof. This is true and elementary, but please add a sentence explaining that f(π(p))=0 in A exactly means p_n | f(n) for all but finitely many n, and that over the field Q the integral closure C_A coincides with the algebraic closure of Q in A. Without this, the claim that the conjecture is settled rests on an unstated argument.
- [Introduction, definition of A] The notation for the quotient of the direct product by the direct sum appears in the typeset version as a symbol that reads like 'M_p'; please ensure the intended direct-sum symbol (\bigoplus) is used in the final version.
- [Use of AI] The disclosure is unusually explicit. Since the author states that the argument was independently reconstructed and verified, I do not regard the AI use as a scientific or integrity concern.
Circularity Check
No significant circularity: the theorem is proved from independent external results; the cited conjecture is the target, not an input.
full rationale
Theorem 1.1 is proved by a self-contained contradiction argument relying on the Maynard–Tao theorem [3, Theorem 1.1] and the prime number theorem. The earlier paper [2] is cited only for background and motivation (the conjecture, Luca–Zudilin's criterion, and examples); none of its results are used in the proof of Theorem 1.1. The key step (2.2) is a linear system with k+1 variables and k equations, so a nontrivial integer solution exists by counting; it is not fitted to any data or prediction. The geometric-series manipulation is exact algebra, and the conclusion Φ(n,p_n)=0 for all sufficiently large n in the infinite set M contradicts the PNT asymptotic (2.3). No definition, fitted parameter, or self-citation is equivalent to the claimed output. The self-citations in the introduction are purely contextual, not load-bearing.
Assumptions & free parameters
assumptions (3)
- standard math Prime number theorem: p_n ~ n log n
- standard math Maynard-Tao bounded-gap theorem: for every m, liminf_{n->infinity}(p_{n+m}-p_n)<infinity
- standard math Homogeneous integer linear systems with more variables than equations have nontrivial integer solutions
Cite this review
Pith. "Pith review of On the $\mathcal{A}$-transcendence of a Champernowne-type constant." pith.science (2026). https://pith.science/paper/MT2BBU26
@misc{pith2026260719337,
author = {Pith},
title = {Pith review of: On the $\mathcalA$-transcendence of a Champernowne-type constant},
year = {2026},
howpublished = {\url{https://pith.science/paper/MT2BBU26}},
note = {Machine review of arXiv:2607.19337}
}
abstract
Let $p_n$ denote the $n$-th prime. We prove that for every nonzero polynomial $f(x)\in\mathbb{Z}[x]$, there exist infinitely many positive integers $n$ such that $p_n\nmid f(n)$.
Reference graph
Works this paper leans on
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[2]
Matsusaka, S
T. Matsusaka, S. Seki,Some results on naive transcendence in the ring of integers modulo infinitely large primes, to appear in J. Aust. Math. Soc
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[1]
F. Luca, W. Zudilin,Poor man ’s transcendence for Frobenius traces of elliptic curves, to appear in Advanced Studies in Pure Math
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[3]
Maynard,Small gaps between primes, Ann
J. Maynard,Small gaps between primes, Ann. of Math.181(2015), 383–413
2015
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[4]
Rosen,A finite analogue of the ring of algebraic numbers, J
J. Rosen,A finite analogue of the ring of algebraic numbers, J. Number Theory208(2020), 59–71. Nagahama Institute of Bio-Science and Technology, 1266, Tamura, Nagahama, Shiga, 526-0829, Japan Email address:s seki@nagahama-i-bio.ac.jp 4
2020
Reviewed August 1, 2026 · model on record in the stance chip above.
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