REVIEW 4 minor 14 references
Prime-Interval Algebras
T0 review · 0 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash
Pith's one-line read From any positive integer n, the paper builds a finite quotient ring and one polynomial remainder whose nonzero monomial degrees are exactly the primes in (n,2n]; if n is prime, the smallest such degree is the next prime.
desk verdict Correct, modest, and honestly positioned algebraic encoding of interval primes; the support-expansion theorem is real and the proof holds; worth referee time. 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 central object is the dyadic modulus R_n, the squarefree product of all primes in (n,2n], realized as a gcd quotient of the central binomial coefficient. The argument runs through the Chinese remainder decomposition Z/R_nZ ≅ ∏ F_p and applies Frobenius in each residue field: in characteristic p, (1+X)^{R_n} = (1+X^p)^{R_n/p}, and because p > n, every term beyond degree p vanishes modulo X^{2n+1}. The leftover is a single monomial X^p with nonzero coefficient, and that coefficient has additive order p modulo R_n. Equivalent annihilator and quotient formulations recover the same primes as jumps in a filtration, as nilpotency indices, and as Jordan-block lengths.
What would settle it
For a single n, say n=1000, compute R_n by the paper's formula and compare it with an independent product of the primes in (1000,2000]; if any prime at most 1000 divides R_n, or any interval prime is missing or appears with multiplicity greater than one, the central identity fails. Equivalently, use Legendre's formula to test whether the claimed inequality v_ℓ(binom(2n,n)) ≤ 2v_ℓ(n!) holds for all primes ℓ ≤ n; a single prime violating it would falsify Proposition 2.1.
Extended reading notes
Core claim
The paper proves that for every n ≥ 1 the integer R_n = binom(2n,n)/gcd(binom(2n,n),(n!)^2) equals the squarefree product of the primes strictly between n and 2n. In the quotient ring B_n = (Z/R_nZ)[X]/(X^{2n+1}), the single modular exponentiation f_n = (1+X)^{R_n}-1 expands as Σ_{n<p≤2n} (R_n/p) X^p, so the support of f_n is exactly that prime set. Each coefficient R_n/p has additive order p, giving an independent certificate of the corresponding prime. Consequently the least nonzero support degree is the least prime greater than n; if n=p_k is prime, that degree is p_{k+1}. The paper also derives the same extraction from an annihilator filtration and extends the support theorem to interval
Load-bearing premise
The whole construction rests on the inequality that every prime ℓ ≤ n appears in the central binomial coefficient at most twice as often as it appears in n!; if this ever failed, small-prime factors would survive the gcd, R_n would cease to be squarefree, and the monomial support theorem would collapse.
Editorial extensions
If this is right
- For every n, reading the support of f_n gives the complete set of primes in (n,2n] without enumerating smaller primes or factoring R_n.
- When n is prime, the smallest support degree is the next prime p_{k+1}, so the successor prime follows directly from the preceding prime.
- Each nonzero coefficient R_n/p has additive order p, so coefficient arithmetic alone certifies which prime each monomial represents.
- The annihilator filtration of f_n records the interval primes as its jumps, and the differences between jump locations are the corresponding prime gaps in reverse order.
- The construction extends to intervals (a,b] with b ≤ 2a, replacing R_n by a gcd quotient of b!/a! and recovering the primes in that interval.
Reading between the lines
- Inference: Iterating the least-support extraction starting from n=2 would generate the sequence of primes by repeatedly applying the construction to the previous prime; this is implied by the successor-prime corollary but is not developed as an explicit algorithm.
- Inference: Because R_n is exactly the product of the interval primes, computing R_n for arbitrary n and comparing it with an independent sieve table would give a direct empirical check of the valuation bound, since any surviving small prime would make R_n nonsquarefree.
- Inference: The condition b ≤ 2a appears to mark the boundary of the method; for longer intervals a prime in (a,b] could appear more than once in b!/a!, so testing whether a modified gcd quotient can handle wider intervals is a natural next step.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper defines, for each positive integer n, the modulus R_n = binom(2n,n)/gcd(binom(2n,n),(n!)^2). Proposition 2.1 proves, by Legendre's formula, that R_n is exactly the squarefree product of the primes in (n,2n]. The paper then forms the quotient ring B_n = (Z/R_nZ)[X]/(X^{2n+1}) and the element f_n = (1+X)^{R_n}-1. Theorem 3.2 shows, via the CRT decomposition into F_p-components and Frobenius, that f_n = \sum_{n<p\le 2n} (R_n/p) X^p, so the monomial support of f_n is precisely the set of interval primes, and each coefficient has additive order p. Consequences include the recovery of the least prime greater than n (and hence of p_{k+1} when n=p_k), an annihilator filtration of B_n, quotient and Jordan-block interpretations, an extension to intervals (a,b] with b\le 2a, a layered regrading of discarded Frobenius terms, and a SageMath implementation. The central derivation is elementary and internally consistent.
Significance. If correct, the paper gives a genuinely parameter-free algebraic encoding of interval primes: no prime table, no primality test, and no pre-selected target primes enter the construction. The key novelty is the two-stage procedure: first a gcd quotient removes all small-prime primary factors, leaving the squarefree interval product as a modulus, and then a single Frobenius expansion places each interval prime at its own monomial degree. The paper also provides reproducible SageMath code and worked examples for both the dyadic and general interval cases, which strengthens the verification of the claims. The result is not deep, but it is a clean and explicit manifestation of a classical divisibility fact, and it should be of interest to readers working on prime formulas, polynomial congruences, or finite-ring encodings of arithmetic data.
minor comments (4)
- [§5, Eq. (3)] The denominator in the definition of R_{a,b} is typeset as '(a!)M', which reads as (a!)·M. From the proof and from Appendix B it is clear the intended expression is (a!)^M. Please fix this typographical ambiguity, since the general-interval theorem depends on the exponent M.
- [§8.4] In the two-layer expansion, the reassembled coefficients 78 and 55 are displayed without explaining the CRT reductions. A short note that 78 ≡ 1 (mod 11), 78 ≡ 0 (mod 13), 55 ≡ 0 (mod 11), and 55 ≡ 3 (mod 13) would make the example self-contained and easier to follow.
- [Appendix B] The code is clear, but the function short_interval_modulus forms a! directly via factorial(a), which is less in the spirit of the dyadic routine that avoids constructing n!. Since the general interval section is only secondary, this is not a correctness issue, but a one-sentence remark on the distinction would be useful.
- [§1.1] The identity R_n = \prod_{n<p\le 2n} p is called 'unexpected'. It follows quickly from Legendre's formula, so the wording is a slight overstatement; consider 'useful' or 'explicit' instead of 'unexpected'.
Circularity Check
No circularity: the central derivation is self-contained, with the product identity proved from Legendre's formula and the Frobenius expansion following from standard finite-field facts.
full rationale
The paper's central claim is that R_n, defined purely as binom(2n,n)/gcd(binom(2n,n),(n!)^2), equals the product of primes in (n,2n], and that f_n = (1+X)^{R_n}-1 has support exactly the interval primes. This is a genuine derivation, not a disguised input. Proposition 2.1 proves the product identity directly using Legendre's formula and the observation that each prime in the interval contributes exactly once while small primes are removed by the gcd; no prime data is fed into the definition of R_n. The Frobenius expansion (Theorem 3.2) follows from the fact that in characteristic p, (1+X)^p = 1+X^p, combined with p > n so higher terms vanish modulo X^{2n+1}. The coefficients R_n/p are then predicted, and their additive orders are computed from the modulus, not fitted. The self-citations [13,14] are explicitly contextual comparisons (and one is by the same first author), but they are not load-bearing: no step in the proof invokes them to establish the main theorem. The general interval extension (Section 5) uses the same mechanism with an explicitly defined R_{a,b}, again with no dependence on prior knowledge of primes in the interval. All supporting computations and the SageMath code are independent checks. There is no fitted parameter, no normalization chosen to force the output, and no uniqueness theorem imported from the authors' prior work. The derivation is self-contained against classical external results (Legendre's formula, CRT, Frobenius), so the circularity score is 0.
Assumptions & free parameters
assumptions (5)
- standard math Legendre's formula for p-adic valuations of n! and binom(2n,n), giving v_ℓ(C) ≤ 2v_ℓ(n!) for ℓ≤n
- standard math Chinese remainder theorem decomposition of Z/R_nZ as a product of F_p
- standard math Frobenius endomorphism in characteristic p: (1+X)^p = 1+X^p in F_p[X]
- standard math Bertrand's postulate: there is at least one prime in (n,2n]
- standard math Swinging-factorial recurrence m! = floor(m/2)!^2 · sf(m)
Cite this review
Pith. "Pith review of Prime-Interval Algebras." pith.science (2026). https://pith.science/paper/5HGXRUOG
@misc{pith2026260722347,
author = {Pith},
title = {Pith review of: Prime-Interval Algebras},
year = {2026},
howpublished = {\url{https://pith.science/paper/5HGXRUOG}},
note = {Machine review of arXiv:2607.22347}
}
abstract
Starting from a positive integer $n$ and no a priori information about the primes above it, we construct a polynomial quotient ring that recovers exactly the primes in $(n,2n]$ from a single modular exponentiation. The primes occur simultaneously as the nonzero monomial degrees of the resulting polynomial remainder, and each coefficient independently certifies its corresponding prime through its additive order. When $n=p_k$ is prime, the least nonzero degree is $p_{k+1}$. Thus the next prime is recovered from the preceding prime alone, without using the index $k$, the prime-counting function, a prime table, nor any primality tests. We develop the underlying ring structure, give equivalent annihilator and quotient formulations, extend the result to shorter intervals, and provide a SageMath implementation.
Reference graph
Works this paper leans on
-
[2]
N. Altafi, A. Iarrobino, P. Macias Marques,Jordan type of an Artinian algebra, a survey, in Lefschetz Properties: Current and New Directions, Springer INdAM Ser.59, Springer, Singapore, 2024, 1–27. https://doi.org/10.1007/978-981-97-3886-1_1
-
[3]
A. O. L. Atkin, D. J. Bernstein,Prime sieves using binary quadratic forms, Math. Comp.73(2004), 1023–1030
2004
-
[4]
C. Bays, R. H. Hudson,The segmented sieve of Eratosthenes and primes in arithmetic progressions to 1012, BIT17(1977), 121–127
1977
-
[5]
R. P. Brent, P. Zimmermann,Modern Computer Arithmetic, Cambridge Monographs on Computational and Applied Mathematics 18, Cambridge University Press, Cambridge, 2010
2010
-
[6]
G. H. Hardy, E. M. Wright,An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, Oxford, 2008
2008
-
[7]
Matomäki,Prime-representing functions, Acta Math
K. Matomäki,Prime-representing functions, Acta Math. Hungar.128(2010), 307–314. https://doi.org/10. 1007/s10474-010-9191-x
2010
-
[8]
W. H. Mills,A prime-representing function, Bull. Amer. Math. Soc.53(1947), 604. https://doi.org/10. 1090/S0002-9904-1947-08849-2
1947
-
[9]
OEIS Foundation Inc.,The On-Line Encyclopedia of Integer Sequences, A261130: Product of primesp withn < p≤2n, https://oeis.org/A261130
Show all 14 references
-
[10]
OEIS Foundation Inc.,The On-Line Encyclopedia of Integer Sequences, A056040: Swinging factorial, https://oeis.org/A056040. [11]C. Pomerance,Divisors of the middle binomial coefficient, Amer. Math. Monthly122(2015), 636–644. [12]P. Pritchard,Linear prime-number sieves: a family...
2015
-
[13]
Prunescu, J
M. Prunescu, J. M. Shunia,On arithmetic terms expressing the prime-counting function and then-th prime, arXiv:2412.14594, 2024; revised 2025. https://arxiv.org/abs/2412.14594
2024 arXiv
-
[14]
Prunescu, J
M. Prunescu, J. M. Shunia,Elementary closed-forms for non-trivial divisors, arXiv:2510.26939, 2025. https://arxiv.org/abs/2510.26939
2025
-
[15]
The Sage Developers,SageMath, the Sage Mathematics Software System, Version 10.9, 2026, https: //www.sagemath.org
2026
-
[16]
The Sage Developers,SageMath Reference Manual: Univariate Polynomial Quotient Rings, Version 10.9 documentation, https://doc.sagemath.org/html/en/reference/polynomial_rings/
-
[17]
E. M. Wright,A prime-representing function, Amer. Math. Monthly58(1951), 616–618. https://doi.org/ 10.2307/2306356. 15
1951 doi
Reviewed August 1, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.