{"id":"9627427b-acaf-4d23-90af-9417126ca124","arxiv_id":"2506.21642","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every base b≥2, infinitely many primes have a reversed digit string with at most Ωb prime factors, with explicit Ωb given.","lead":"This paper proves that in any base b, infinitely many primes have a reversed digit string that is an almost prime, with an explicit bound on the number of prime factors. It establishes a Bombieri-Vinogradov type equidistribution theorem for reversed primes and uses sieve methods to count them.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The printed Ωb values for 3≤b≤10 rest on Table 14.1's κb, which is justified only by 'a collection of plots'; a wrong κb would invalidate those explicit constants, though an analytic fallback preserves existence of some finite Ωb.","rationale":"The paper is a serious analytic number theory proof whose central reduction, sieve applications, and Type I/II exponential sum estimates appear structurally sound and free of circularity. The constants in the chain ξ0(b) → ιb → κb are the only place where the printed output depends on unverified numerical information: Table 14.1 asserts κb for 3 ≤ b ≤ 10 by plots, and the explicit Ωb values in Table 1.1 are monotone in κb, so an under-estimated κb would make those Ωb inadmissible. The analytic fallback (14.1) gives an explicit finite κb for every b, so the main existence theorem and Theorem 1.6/1.3 are not endangered; only the displayed small-base constants are at risk. This fully matches the reader's weakest-assumption analysis, and the proposed interval-arithmetic check would settle it. I therefore keep the reader's CONDITIONAL verdict unchanged.","tokens_in":73691,"tokens_out":15094,"duration_ms":159064,"concrete_test":"Implement T_{b,κ}(α) from (6.31) using interval arithmetic or a certified global-optimization routine for b = 3, ..., 10, and verify for each b that the threshold inequality equivalent to (11.5) fails at κ = κb − 1 and holds at κ = κb, where the threshold is max T_{b,κ}(α) < 1/(b max T_{b,1}(α)). This directly validates Table 14.1 and hence the Ωb column of Table 1.1; also recompute ηb from (6.24) and the ξ0(b) values in Table 14.2 to the displayed precision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The explicit constants in Theorem 1.1 flow from (15.9), Ωb = 1 + ⌈1/ξ0(b)⌉, where ξ0(b) is defined in (13.1). One term of (13.1) depends on ιb, hence on κb, the integer defined in (11.6) as the minimal κ satisfying b^{-ζb,1} b^{1-ζb,κ} < 1. For b = 3, ..., 10, Section 14.1 gives κb only via 'checking numerically' and 'a collection of plots' (Table 14.1); no code, interval bounds, or certified error estimates are supplied. The b = 2 case has a 13-interval grid check, but even there the margin on the first interval is extremely small, and no comparable certificate is provided for b ≥ 3. If any tabulated κb is too small, the corresponding ξ0(b) is too large and the displayed Ωb in Table 1.1 is not admissible. This does not destroy the structural theorem: the analytic bound (14.1) supplies an explicit, much larger κb for every b, so the existence of some finite admissible Ωb and the Bombieri–Vinogradov estimate survive. The concern is therefore specifically about the rigor and reproducibility of the numerical certificate supporting the printed constants, which is exactly the point a conditional verdict should hinge on.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":73912,"tokens_out":9049,"duration_ms":107820,"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":[{"comment":"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.","section":"§14.1, Table 14.1; equations (11.6), (11.14), (13.1), (15.9)"},{"comment":"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.","section":"§6.15 and §14.1 (finite numerical checks for 2 ≤ b ≤ 4)"}],"minor_comments":[{"comment":"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.","section":"§15.6, around (15.7)"},{"comment":"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.","section":"Table 14.2"}],"recommendation":"major_revision","confidential_remarks":"I believe the central mathematics is likely correct and the structural theorem survives even if the small-base table is replaced by the analytic bound. The revision should focus on the numerical certificate for κ_b; without that, the advertised constants in Table 1.1 are not established. This is a fixable gap rather than a reason to reject."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main thing to know: this paper appears to deliver the first proof in every base b≥2 that infinitely many primes have an almost-prime reverse, with explicit Ω_b. It gets there through a Bombieri-Vinogradov type theorem (Thm 1.3) for the distribution of reverses of primes in arithmetic progressions, proved via a long Type I/Type II exponential sum analysis in the Mauduit-Rivat tradition, then linear and weighted sieves. The structure is coherent: no fitted parameter enters the target; κ_b, ξ_0(b), Ω_b are outputs of the argument. The paper is explicit about constants and gives Table 1.1 for 2≤b≤10.\n\nWhat it does well: the individual progression estimates (Thm 1.4) extend recent work of Bhowmik-Suzuki and Chourasiya-Johnston from huge bases to all b≥2, and the averaged estimate (Thm 1.6/1.3) is the key new engine. I did not machine-check the deep estimates, but the derivation is detailed and I see no internal contradiction. Dependencies on the authors' own prior work (Mauduit-Rivat lemmas, base-2 reversible primes) are legitimate and explicit.\n\nWhere the soft spots are: Section 14.1 fixes κ_b for 3≤b≤10 by 'checking numerically' on a finite grid and by plots. No code, no interval bounds. These κ_b feed directly into ξ_0(b) and thus the printed Ω_b in Table 1.1. If a tabulated κ_b is too small, the corresponding ξ_0(b) and Ω_b are not admissible. For b=2 there is a 13-interval grid check, but the margin on the first interval is quite small; for b≥3 it's just plots. This does not kill the structural theorem: the analytic bound (14.1) produces an explicit, larger κ_b for every b, so existence of some finite Ω_b survives. The concern is specifically the rigor and reproducibility of the displayed constants.\n\nBottom line: this is a serious paper for analytic number theorists working on digits and sieves. It deserves a full peer review. I'd want the referee report to ask for a certified computation of κ_b, or at least a clear statement that the rigorous constants come from the analytic bound while Table 14.1 is heuristic. That is an amendable weakness, not a fatal one. I'd bring it to reading group and would cite it.","headline":"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.","tokens_in":74527,"tokens_out":1807,"would_cite":true,"duration_ms":21837,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A63","11N05","11N36"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $\\Omega_2 = 228$.","keywords":["prime numbers","almost primes","reversed base-b expansion","Bombieri-Vinogradov theorem","exponential sums","sieve methods","digit reversal","Dirichlet kernel"],"falsifier":"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.","tokens_in":73438,"feed_emoji":"🔢","tokens_out":7519,"duration_ms":85867,"temperature":0.7,"pith_summary":"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.","feed_headline":"Every base has infinitely many primes with almost-prime reverse","feed_subtitle":"For b=2, the reverse has at most 228 prime factors; the proof uses a Bombieri–Vinogradov theorem for reversed primes.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $b \\ge 2$."],"supporting_citations":[{"why":"introduces the almost-prime-replacement paradigm that motivates Theorem 1.1.","marker":"[30]"},{"why":"supplies the van der Corput inequality and the exponential-sum strategy used for the Type II sums.","marker":"[25]"},{"why":"establishes the base-2 reversible-primes result that Theorem 1.1 extends, and provides the Type I treatment.","marker":"[11]"},{"why":"gives the linear sieve bounds used in Lemma 3.1 to detect primes with no small prime factor in the reverse.","marker":"[18]"},{"why":"gives the weighted sieve with Richert weights used in Lemma 3.2 to bound the total number of prime factors.","marker":"[21]"},{"why":"provides the van der Corput variant and several Fourier estimates used throughout Sections 5 and 6.","marker":"[24]"},{"why":"supplies the carry-propagation argument for Type II sums, adapted in Lemma 8.1.","marker":"[26]"}],"fun_headline_variants":["Almost-prime reverses exist in every base","In any base, many primes have almost-prime reverses","For every base, reverse of some primes is almost prime","Bombieri–Vinogradov for reversed primes yields almost-prime reverses","Many primes in every base have almost-prime reverses"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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 $b$.","fun_headline_variants_meta":{"raw":{"variants":["Almost-prime reverses exist in every base","In any base, many primes have almost-prime reverses","For every base, reverse of some primes is almost prime","Bombieri–Vinogradov for reversed primes yields almost-prime reverses","Many primes in every base have almost-prime reverses"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000649,"raw_usage":{"total_tokens":2956,"prompt_tokens":902,"completion_tokens":2054,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":518,"completion_tokens_details":{"reasoning_tokens":1973}},"tokens_in":518,"tokens_out":2054,"duration_ms":17164,"temperature":1.0,"reasoning_tokens":1973,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:41:07.521318+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"R´enyi, On the representation of an even number as the sum of a single prime and single almost-prime number , Izv","cited_arxiv_id":null,"evidence_quote":"introduces the almost-prime-replacement paradigm that motivates Theorem 1.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the van der Corput inequality and the exponential-sum strategy used for the Type II sums."},{"cited_title":"Dartyge, B","cited_arxiv_id":null,"evidence_quote":"establishes the base-2 reversible-primes result that Theorem 1.1 extends, and provides the Type I treatment."},{"cited_title":"Friedlander and H","cited_arxiv_id":null,"evidence_quote":"gives the linear sieve bounds used in Lemma 3.1 to detect primes with no small prime factor in the reverse."},{"cited_title":"Halberstam and H.-R","cited_arxiv_id":null,"evidence_quote":"gives the weighted sieve with Richert weights used in Lemma 3.2 to bound the total number of prime factors."},{"cited_title":"Mauduit and J","cited_arxiv_id":null,"evidence_quote":"provides the van der Corput variant and several Fourier estimates used throughout Sections 5 and 6."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the carry-propagation argument for Type II sums, adapted in Lemma 8.1."}],"review_version":1}