{"id":"48bd64fb-d79f-43c0-bb7b-70e58467019f","arxiv_id":"2607.19337","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For every nonzero f in Z[x], infinitely many primes p_n fail to divide f(n), settling Conjecture 6.2 of Matsusaka and Seki.","lead":"This paper proves that for any nonzero polynomial with integer coefficients, infinitely many prime-indexed primes p_n fail to divide the polynomial's value at n. This settles a conjecture about the transcendence of a Champernowne-type constant in the ring of integers modulo all primes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The reader's ACCEPT verdict is justified. The stress-test pass found no mathematical flaw in the proof of Theorem 1.1. The argument is elementary after invoking Maynard–Tao and PNT, both correctly applied. The only unproved assertion is the equivalence to [2, Conjecture 6.2], which is a comparative claim about the strength of the theorem relative to the conjecture. Since Theorem 1.1 is strictly stronger than the negation of the conjecture, the equivalence is not necessary for settling the conjecture. The reader's weakest assumption (Maynard–Tao) is a genuine external dependency but a standard theorem, and no uniformity beyond its statement is needed. The agreement is partial because the reader flagged Maynard–Tao as a potential weak point, while the stress-test finds it fully adequate; the real (but non-load-bearing) gap is the unproved equivalence.","tokens_in":3574,"tokens_out":18578,"duration_ms":152500,"concrete_test":"Verify the claimed equivalence between Theorem 1.1 and [2, Conjecture 6.2] by deriving the converse direction (i.e., that failure of the conjecture implies only finitely many n with p_n∤f(n)); if the converse fails, the paper overclaims but the proof of Theorem 1.1 still stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 is internally consistent and correct. The application of Maynard–Tao [3, Theorem 1.1] to obtain an infinite set M with a fixed tuple of offsets is valid; the pigeonhole step does not require extra uniformity. The construction of the nonzero polynomial Φ and the nontrivial integer solution c_j is sound. The quotient Φ(n,p_n)/ρ(p_n) is exactly an integer for n∈M, and the geometric-series computation shows it is o(1), forcing it to vanish for all large n∈M. This contradicts the PNT-based asymptotic (2.3). The only unproved assertion is the claimed equivalence to [2, Conjecture 6.2] in the Introduction; since Theorem 1.1 implies the conjecture via the natural translation f(π(p))=0, this peripheral issue does not affect the central theorem. No load-bearing defect found.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":3721,"tokens_out":10838,"duration_ms":105718,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"Introduction, last paragraph"},{"comment":"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.","section":"Introduction, definition of A"},{"comment":"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.","section":"Use of AI"}],"recommendation":"accept","confidential_remarks":"I see no load-bearing defect. The proof depends essentially on the Maynard–Tao theorem, but this is a published result and is used in the correct form. The only substantive request I would make to the authors is to add the elementary justification of the equivalence with Conjecture 6.2; this is a local addition and does not affect the central theorem. I recommend acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the Seki note. The main theorem is correct and it settles the Matsusaka–Seki conjecture. I went through the proof line by line and did not find any load-bearing gap. The translation from A-transcendence to the elementary statement about p_n ∤ f(n) is exactly right—an element of C_A is algebraic over Q, and algebraicity is equivalent to being annihilated by a nonzero integer polynomial, so the conjecture is literally Theorem 1.1.\n\nWhat's new is the method. Using the Maynard–Tao bounded gaps theorem to produce infinitely many blocks of consecutive primes with fixed offsets is a neat trick, and the homogeneous linear system plus geometric-series cancellation is a clean way to force a contradiction. The proof is short but complete. The use of PNT for the nonvanishing asymptotic is standard.\n\nMinor soft spots: the equivalence with Conjecture 6.2 is asserted without proof. It's immediate and the stress-test is right that it's peripheral, but a one-sentence explanation would help. The proof also leans on Maynard–Tao as a black box; if a reader objects to importing a deep sieve theorem for a problem about polynomial divisibility, that's a matter of taste, not correctness. The note is terse and relies on [2] for the framework; new readers will need that paper.\n\nThe AI disclosure is transparent and the author says they reconstructed and verified the argument; the proof itself is fully written, so I don't see an integrity issue.\n\nWorth a serious referee. It's a small, clean result that closes a specific open problem, and the technique could be useful elsewhere. I'd cite it, and I might bring it to a reading group as a compact example of a deep theorem applied to a simple question.","headline":"Correct proof of the Matsusaka–Seki conjecture, with a surprising Maynard–Tao input; minor presentation issues but no mathematical gaps.","tokens_in":4161,"tokens_out":7094,"would_cite":true,"duration_ms":61791,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11J81","11A41","11N05","11C08"],"pacs":[],"model":"deepseek-v4-flash","headline":"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).","keywords":["Champernowne constant","A-transcendence","integral closure","prime divisibility","bounded gaps","polynomial values","prime indexing","transcendence"],"falsifier":"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.","tokens_in":3459,"feed_emoji":"🔢","tokens_out":17343,"duration_ms":143394,"temperature":0.7,"pith_summary":"The paper proves a purely number-theoretic statement: for any nonzero polynomial f with integer coefficients, there are infinitely many positive integers n such that the n-th prime p_n does not divide f(n). This is exactly the claim that the Champernowne-type element π(p) = (n mod p_n)_p is transcendental in the strong sense, i.e., it does not belong to the integral closure of Q in the ring A of prime-indexed residue sequences. The result settles a conjecture from earlier work of the same authors, which had only been established for polynomials of degree at most two or under auxiliary conjectures. The proof is short: it combines the bounded-gaps theorem for primes with a cancellation identity to show that if such a polynomial existed, a nonzero two-variable polynomial would have to vanish on an infinite set of pairs (n,p_n), contradicting its asymptotic growth.","feed_headline":"For every polynomial, primes fail to divide values infinitely often","feed_subtitle":"Settles the conjecture that the Champernowne-type constant is not algebraic over Q.","key_machinery":"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 → ∞.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["For any polynomial, infinitely many primes refuse to divide its values","Every polynomial has infinitely many prime indices where it's not divisible","No integer polynomial can force divisibility by all its prime indices","Champernowne-type constant shown non-algebraic over rationals","Primes skip dividing polynomial values at infinitely many spots"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["For any polynomial, infinitely many primes refuse to divide its values","Every polynomial has infinitely many prime indices where it's not divisible","No integer polynomial can force divisibility by all its prime indices","Champernowne-type constant shown non-algebraic over rationals","Primes skip dividing polynomial values at infinitely many spots"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000976,"raw_usage":{"total_tokens":3907,"prompt_tokens":595,"completion_tokens":3312,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":339,"completion_tokens_details":{"reasoning_tokens":3225}},"tokens_in":339,"tokens_out":3312,"duration_ms":23350,"temperature":1.0,"reasoning_tokens":3225,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T12:43:06.467689+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}