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Prime numbers with an almost prime reverse

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Pith's one-line read For every integer base $b \ge 2$, infinitely many primes have a reversed base-$b$ expansion that is an almost prime with at most $\Omega_b$ prime factors, and the proof gives explicit values such as $\Omega_2 = 228$.

desk verdict A substantial, carefully executed analytic number theory paper that proves the main existence theorem in every base; the only real weakness is that the printed small-base constants rest on un-certified numerical checks. read the letter →

arxiv 2506.21642 v2 pith:BMDN4XQK submitted 2025-06-25 math.NT

Cécile Dartyge , Joël Rivat , Cathy Swaenepoel This is my paper · ORCID
classification math.NT MSC 11A6311N0511N36
keywords primenumbersalmostprimesreversedbase-bexpansionBombieri-VinogradovtheoremexponentialsumssievemethodsdigitreversalDirichletkernel
topics The Twin Prime Conjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that in every integer base $b \ge 2$ there are infinitely many primes whose reversed base-$b$ expansion is an almost prime, meaning a number with a bounded number of prime factors. The quantitative form is a lower bound $\gg b^{\lambda}/\lambda^2$ for the number of $\lambda$-digit primes $p$ with $\Omega(R_\lambda(p)) \le \Omega_b$, and the proof supplies explicit values such as $\Omega_2 = 228$. The engine is a Bombieri-Vinogradov type theorem: reversed primes are well distributed in arithmetic progressions modulo $d$, on average over $d$ up to $b^{\xi\lambda}$ with $d$ coprime to $b(b^2-1)$. That average distribution feeds into linear and weighted sieves to detect primes whose reverse has no small prime factor and few prime factors in total. If correct, this answers a natural digit-analogue of the classical almost-prime replacement strategy in every base.

What carries the argument

The load-bearing object is the exponential sum $F_\lambda(\alpha, \vartheta) = b^{-\lambda}\sum_{0 \le n < b^{\lambda}} e(\alpha R_\lambda(n) - \vartheta n)$, together with its product formula $|F_\lambda(\alpha, \vartheta)| = \prod_{j=0}^{\lambda-1} |K_b(\alpha b^{\lambda-1-j} - \vartheta b^j)|$, where $K_b(x) = \sin(\pi b x)/(b \sin \pi x)$ is the normalized Dirichlet kernel. This factorization lets the proof separate the reversed-digit variable $\alpha$ from the natural-digit variable $\vartheta$ and transfer estimates between them; repeated H\"older and Cauchy-Schwarz steps, combined with $L^\kappa$ norm bounds on products of these kernels, control the Type I and Type II sums arising from Vaughan's identity. A second mechanism is the carry-propagation lemma: for most pairs $(m,n)$ the difference $R_\lambda(m(n+r)) - R_\lambda(mn)$ is determined by low-order digits because the higher digits are all $b-1$, which is what makes the Type II analysis tractable.

What would settle it

Recompute $\kappa_2, \ldots, \kappa_{10}$ by rigorous interval arithmetic from the definition (11.6), verifying $b^{-\zeta_{b,1}} b^{1-\zeta_{b,\kappa_b}} < 1$ for the claimed $\kappa_b$ and failing for each smaller integer; if Table 14.1 is not reproduced, the explicit $\Omega_b$ in Table 1.1 are not established, whereas if it is reproduced, the numerical step in the proof is confirmed.

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Extended reading notes

Core claim

The central discovery is that the arithmetic obstruction to reversing the digits of primes is mild enough to be handled by averaging. For primes $p$ in a $\lambda$-digit interval, the count of $p$ with $R_\lambda(p) \equiv a \bmod d$ is $\pi_\lambda(t)/d$ plus an error that, summed over all $d \le b^{\xi\lambda}$ with $\gcd(d, b(b^2-1)) = 1$ and all residue classes, is $\ll b^{\lambda - c\sqrt{\lambda}}$. Once this Bombieri-Vinogradov statement is in place, the linear and weighted sieves give Theorem 1.1. In addition the paper obtains an upper bound of the same shape $\ll b^{\lambda}/\lambda^2$ for primes whose reverse is itself prime, and a Siegel-Walfisz-type range $d \le \exp(c\sqrt{\lambda})$ in which the asymptotic holds for individual moduli.

Load-bearing premise

The load-bearing premise is that the finite numerical checks in Section 14.1 correctly identify $\kappa_b$ for $2 \le b \le 10$; those values feed directly into $\xi_0(b)$ and therefore into the explicit $\Omega_b$, and a mistake there would change the constants even though the analytic bound (14.1) would still supply some finite $\Omega_b$ for every $b$.

Editorial extensions

If this is right

  • For any $b \ge 2$ there are infinitely many primes whose reverse has at most $\Omega_b$ prime factors, with explicit constants such as $\Omega_2 = 228$, $\Omega_3 = 333$, and $\Omega_{10} = 1378$.
  • The same sieve framework shows that primes whose reverse is also prime are rare in the expected sense: their number is $\ll b^{\lambda}/\lambda^2$, matching the conjectured order of magnitude up to the constant.
  • As the base grows, the admissible $\Omega_b$ is $O(b^2)$ with an explicit leading constant $538.106849\ldots$, so the quality of the method degrades polynomially in $b$.
  • The average distribution statement (Theorem 1.3) is a standalone Bombieri-Vinogradov theorem for reversed primes and can be reused in any sieve problem whose sequence is obtained by digit reversal.
  • A byproduct is a Siegel-Walfisz-type asymptotic for primes with squarefree reverse, valid for all bases $b \ge 2$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The explicit $\kappa_b$ values in Table 14.1 are likely improvable: sharper numerical optimization of $T_{b,\kappa}$ or a better analytic bound would lower $\xi_0(b)$ and, through $\Omega_b = 1 + \lceil 1/\xi_0(b)\rceil$, would give smaller admissible almost-prime bounds.
  • The product-formula method is not tied specifically to digit reversal; the same $F_\lambda$ machinery should apply to any digit operation that factors digitwise, such as complementation or reversal followed by a fixed affine map.
  • The proof's finite numerical step is the part a reader should automate first: replacing the reported finite-grid checks and plots by certified interval arithmetic would remove the only non-rigorous-looking step from the explicit-constants argument.
  • The squarefree-reverse asymptotic suggests that fully quantitative counts of primes whose reverse is $r$-free, for fixed $r$, could be pushed further in bases where the involved constants are verified rigorously.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper studies, for every integer base b ≥ 2, the reversal R_λ(n) of the λ-digit base-b expansion of n. Its main result (Theorem 1.1) asserts that for every b there is an explicit Ω_b such that the number of primes p in [b^{λ−1}, b^λ) with Ω(R_λ(p)) ≤ Ω_b is ≫ b^λ/λ², and it tabulates admissible Ω_b for 2 ≤ b ≤ 10. The central new input is Theorem 1.3, a Bombieri–Vinogradov-type estimate for the sequence R_λ(p) in arithmetic progressions modulo d with gcd(d, b(b²−1)) = 1. The proof is carried out through a Vaughan decomposition into Type I and Type II sums, detailed exponential-sum estimates for Λ(n)e(hR_λ(n)/d), and explicit L^κ and mean-value bounds for the generating function F_λ. The paper also proves an individual distribution theorem (Theorem 1.4), a Siegel–Walfisz-type corollary (Theorem 1.5), and an upper bound for primes whose reversal is prime (Theorem 1.2). A key intermediate parameter κ_b, defined by an inequality in §11, is evaluated numerically for small bases in §14.1, and these values feed directly into the displayed constants ξ_0(b) and Ω_b.

Significance. If the printed constants are fully certified, this is a substantial unconditional advance: it extends the base-2 reversible-prime result to every base, removes the large-base restriction in the recent work of Bhowmik–Suzuki and Chourasiya–Johnston, and gives a fully explicit lower-bound density for primes whose reversal is an almost prime. The proof is distinguished by its level of explicitness: constants are tracked throughout, no fitted free parameter enters the target estimates, and several auxiliary Fourier/moment lemmas are stated in a reusable form. The principal weakness is not structural but computational: the numerical certificate for κ_b for 3 ≤ b ≤ 10 is not reproducible as printed. Since the analytic bound (14.1) supplies an explicit admissible κ_b for every b, the existence of some finite explicit Ω_b survives even if the small-base table were wrong; however, the advertised Table 1.1 values would not be established in their current form.

major comments (2)
  1. [§14.1, Table 14.1; equations (11.6), (11.14), (13.1), (15.9)] The proof of the displayed values κ_b for 3 ≤ b ≤ 10 rests on the statements 'checking numerically' and 'a collection of plots', with no code, interval arithmetic, or certified error bounds. These values are load-bearing: κ_b enters ι_b via (11.14), ξ_0(b) via (13.1), and hence Ω_b via (15.9); if any tabulated κ_b is too small, the printed Ω_b in Table 1.1 is not admissible. The analytic upper bound (14.1) provides an explicit, albeit much larger, admissible κ_b for every b, so the structural theorem is not in danger; nevertheless, the numerical certificate for the small-base table should be made rigorous, or the table should be recomputed from the analytic bound. A short interval-arithmetic appendix, or a clearly specified verifiable computation, would resolve this.
  2. [§6.15 and §14.1 (finite numerical checks for 2 ≤ b ≤ 4)] The same certification concern applies, at a smaller scale, to the 'elementary numerical computations' used in Lemma 6.15 to verify (6.22) for 2 ≤ b ≤ 4 and to the base-2 grid check in §14.1. These checks are finite and probably correct, but they are not presented in a form that a referee can verify. Since the affected quantities η_b and κ_b influence the final constants through (6.24) and (11.14), the manuscript should either give a fully specified finite procedure with rigorous error bounds or cite a verifiable computer-assisted proof.
minor comments (2)
  1. [§15.6, around (15.7)] There appears to be an off-by-one error in the intermediate weaker version of Theorem 1.1. From Ω(n) < ξ^{−1} one obtains Ω(n) ≤ ⌈ξ^{−1}⌉ − 1, and with the chosen ξ satisfying eΩ_b < ξ^{−1} < eΩ_b + 1 this gives Ω(n) ≤ eΩ_b. As printed, (15.7) uses ⌊ξ^{−1}⌋ − 1, and the following line asserts ⌊ξ^{−1}⌋ = eΩ_b + 1, which is inconsistent with ξ > (eΩ_b + 1)^{−1}. The final proof via the weighted sieve is unaffected, but this intermediate passage should be corrected.
  2. [Table 14.2] The displayed decimal values of ξ_0(b) are followed by ellipses but without any statement of how they were computed or whether they are rounded or truncated. Since Ω_b is obtained as 1 + ⌈1/ξ_0(b)⌉, a precise convention for these decimal approximations would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is derived from an independent exponential-sum and sieve argument, with κb a defined threshold rather than a fitted input.

full rationale

The paper establishes Theorem 1.1 by an explicit sieve argument whose only analytic input is the averaged distribution result Theorem 1.3. That result is proved from Theorem 1.6 by partial summation (Section 15.1), and Theorem 1.6 is proved from first principles: Vaughan's identity (Lemma 4.1/4.2), an average Type II bound (Lemma 11.1), and an average Type I bound (Lemma 12.2). The target quantity — the set of primes p in a b-adic interval with Ω(Rλ(p)) ≤ Ωb — is never used as an input. The sieve error terms are bounded by the exponential-sum estimates, and Ωb is computed from ξ0(b) via (15.9), so it is an output of the proof, not a fit to the conclusion. The only potentially fragile step is the numerical certification of κb for 2 ≤ b ≤ 10 in Section 14.1, justified by 'checking numerically' and 'a collection of plots' (Table 14.1). This is a reproducibility and rigor concern about the printed constants, but it is not circular: κb is defined as the minimal integer satisfying the inequality b^{-ζb,1} b^{1-ζb,κ} < 1, and the analytic bound (14.1) independently guarantees existence of some admissible κb for every b, so the structural theorem and the Bombieri–Vinogradov estimate survive even if a tabulated value were wrong. Citations to prior work by the same authors, notably [11] and the Mauduit–Rivat lemmas, supply methodology and external analytic tools rather than assuming the target conclusion. No equation in the paper reduces to its own input, and no fitted parameter is renamed as a prediction. Therefore the derivation is self-contained and no circularity is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central argument rests on a small set of standard analytic number theory tools and on its own explicit estimates; no additional axioms, fitted parameters, or invented entities are introduced. The only non-standard numerical input is the finite verification of κb flagged in red_flags.

assumptions (6)
  • standard math Prime Number Theorem with standard error term
    Used at (15.5) to estimate the number of primes in each subinterval P_{λ,i}; standard unconditional theorem.
  • standard math Mertens estimates for products over primes
    Used at (3.13) to show the sieve density V(z) behaves like e^{-γ}/log z times a base-dependent constant; standard.
  • standard math Linear sieve fundamental lemma
    Lemma 3.1, cited to Friedlander-Iwaniec and Iwaniec; gives upper and lower bounds for Θ_i(λ,z).
  • standard math Richert weighted sieve
    Lemma 3.2, cited to Halberstam-Richert and other standard references; converts equidistribution into a count with a bounded number of prime factors.
  • standard math Vaughan's identity
    Lemma 4.1, cited to Iwaniec-Kowalski; decomposes von Mangoldt sums into Type I and Type II sums.
  • standard math Standard harmonic analysis inequalities
    Large sieve, van der Corput, Sobolev-Gallagher, and Bernstein-Zygmund inequalities, used throughout Sections 5-9; all are standard background.

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Pith. "Pith review of Prime numbers with an almost prime reverse." pith.science (2026). https://pith.science/paper/BMDN4XQK

@misc{pith2026250621642,
  author       = {Pith},
  title        = {Pith review of: Prime numbers with an almost prime reverse},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BMDN4XQK}},
  note         = {Machine review of arXiv:2506.21642}
}
abstract

Let $b$ be an integer greater than or equal to $2$. For any integer $n\in \left[b^{\lambda-1}, b^{\lambda}-1\right]$, we denote by $R_\lambda (n)$ the reverse of $n$ in base $b$, obtained by reversing the order of the digits of $n$. We establish a Bombieri-Vinogradov type theorem for the set of the reverses of the prime numbers. Combined with sieve methods, this permits us to prove that there exist $\Omega_b\in\mathbb{N}$ and $c_b>0$ such that, for at least $c_b b^{\lambda} \lambda ^{-2}$ primes $p\in \left[b^{\lambda-1}, b^{\lambda}-1\right]$, the reverse $R_\lambda(p)$ has at most $\Omega_b$ prime factors. Some explicit admissible values of $\Omega_b$ are given.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Zsiflaw--Legeis theorem for arbitrary bases

    math.NT 2025-07 accept novelty 6.0 of 10

    The digital reverse of primes is equidistributed in arithmetic progressions for every base g>=2, with a quantitative error term.

Reference graph

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