{"id":"45a403e3-19bb-40d3-9840-3c978179c7c6","arxiv_id":"2506.12047","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"An explicit arithmetic progression is constructed whose terms are neither prime-plus-Fibonacci nor prime-plus-Lucas numbers.","lead":"This paper proves that there is an endlessly repeating pattern, an arithmetic progression, of whole numbers that can never be written as a prime plus a Fibonacci number nor as a prime plus a Lucas number. It matters as a new existence result in additive number theory, where such patterns were previously found for each family separately but not for both at once.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.1's two covering systems are asserted by \"calculation\" with no certificate; the proof of Theorem 1.1 collapses at (3.2)/(3.3) if either misses one residue, so direct enumeration is the decisive check.","rationale":"The paper's main theorem is an explicit construction, and the construction is valid only if the two covering systems cover all residues. The reader's conditional verdict rests on exactly this. My own preliminary check of the 2- and 3-adic residue structure found no uncovered residue class, so the concern is not that the lemma is false, but that the proof delegates a load-bearing finite computation to the reader. Since the check is simple and decisive, CONDITIONAL is the right status; if the enumeration passes, the theorem should be accepted modulo checking the similarly unverified congruence residues in the display after Lemma 2.1 and the CRT consistency of M and N. I agree with the reader's weakest-assumption identification. No adjustment to the verdict is needed.","tokens_in":8820,"tokens_out":27603,"duration_ms":229634,"concrete_test":"Run a direct enumeration: for each r in 0..1151 check that some of the 33 pairs (a_i,m_i) satisfies r ≡ a_i (mod m_i), and for each s in 0..575 check that some of the 15 pairs (b_j,n_j) satisfies s ≡ b_j (mod n_j). The proof is sound at this step iff both unions have full size (1152 and 576). This can be done in a few lines in any CAS; no large integers are involved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Lemma 2.1. The proof says only \"By calculation, one can easily verify Lemma 2.1.\" Both systems are used to guarantee that, for every Fibonacci index m and every Lucas index n, there is an i0 or j0 with m ≡ a_i0 (mod m_i0) and n ≡ b_j0 (mod n_j0); this is what restricts p and q to the finite lists in (3.2) and (3.3). If either system fails to cover even one residue modulo 1152 (resp. 576), there is a possible representation x = p + F_m (or x = q + L_n) that the contradiction tables never address, and Theorem 1.1 is unproved. No code, certificate, or enumeration is included. A hand decomposition of the two systems into their 2-adic and 3-adic components suggests the coverage is present, so I do not claim a counterexample; the gap is that the decisive computational assertion is not independently verified in the manuscript.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem 1.1: for two explicitly displayed very large integers M and N, every term of the arithmetic progression Mk+N (k≥0) lies in the set B, meaning it is simultaneously not of the form p+F_m for any prime p and Fibonacci number F_m, and not of the form q+L_n for any prime q and Lucas number L_n. The proof constructs a system of 48 congruences (33 for the Fibonacci branch, 15 for the Lucas branch), invokes two covering systems modulo 1152 and 576 to reduce the infinite families of indices to finitely many cases, uses the Chinese remainder theorem to identify the solution set as the progression Mk+N, and then rules out each finite case by tables of F_m and L_n modulo 17, 19, and 47. Corollaries give analogous one-sided progressions for B_f and B_l.","tokens_in":9082,"tokens_out":14634,"duration_ms":148411,"significance":"If fully verified, the result is a clean and nontrivial addition to the Romanoń/de Polignac literature: it gives an explicit arithmetic progression whose terms avoid both p+F_m and q+L_n simultaneously, going beyond the earlier one-sided results of Jones, Ismailescu, Šiurys, and Ismailescu–Shim. The overall proof structure is transparent: once the covering systems and the modular identities are granted, the final contradiction is a finite check using hand-verifiable tables. The paper also states explicit corollaries for the one-sided sets. Its main weakness is that the load-bearing computational material—the covering systems and the many modular identities involving very large primes—is asserted without code, certificate, or reproducible verification, so the result is not machine-checkable as written.","major_comments":[{"comment":"The two covering systems are load-bearing, since they are what restrict the infinitely many Fibonacci and Lucas indices to the finite lists used in (3.2) and (3.3), but the proof is only the sentence 'By calculation, one can easily verify Lemma 2.1.' No certificate is provided for either system. If one residue class modulo 1152 (resp. 576) were missed, the argument would collapse. Please provide an explicit certificate—for example, for each residue class modulo 1152 and 576, list which congruence covers it, or include a short reproducible enumeration script.","section":"Lemma 2.1"},{"comment":"The reduction to p=p_{i_0} and q=q_{j_0} in (3.2) and (3.3) relies on the displayed identities F_{m_i k+a_i} ≡ r_i (mod p_i) for 1≤i≤33, L_{n_j k+b_j} ≡ s_j (mod q_j) for 1≤j≤15, and the equalities m_i=χ_f(p_i), n_j=χ_l(q_j). These are asserted with 'some help of a computer' but no derivation, code, or certificate is included. The argument also requires all the p_i and q_j to be prime, and this is not verified for the very large moduli such as 115561578124838522881 and 1974737795746080149567. Please supply a reproducible verification, ideally a short script or explicit certificates for each congruence and each primality assertion.","section":"Section 3, displayed congruences and equation (3.1)"}],"minor_comments":[{"comment":"The Chinese remainder theorem step silently assumes that the combined system of 48 congruences is consistent. The repeated moduli in the two lists (2, 3, 7, 23, 47, 19, 17, 2207) do have matching residues, but this should be stated explicitly.","section":"Section 3, equation (3.1)"},{"comment":"In a few cases the phrase 'a contradiction to Table X' is compressed: the target residue does appear elsewhere in the table, and the contradiction only follows after restricting to the possible residue classes of the index. This occurs, for example, in the m_i=16 case (2 mod 17 appears in Table 1) and in the m_i=36 and m_i=1152, p_i=1974737795746080149567 cases. Please spell out the residue-class restriction in each such case.","section":"Proof of Theorem 1.1, case analysis"},{"comment":"The corollaries are stated with explicit M and N but no proof is given beyond 'can be deduced.' Since they use different covering systems and different congruences, they should either be proved explicitly or clearly labeled as computational claims requiring the same verification as the main theorem.","section":"Corollaries 4.1–4.3"},{"comment":"There are minor typographical issues: 'inevitablely' in Remark 1.3, 'balence' in Section 3, and inconsistent spacing in the displayed M and N. These do not affect the mathematics.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The core mathematical strategy appears sound and the finite contradiction tables are checkable by hand. My main concern is reproducibility: the two covering systems and the many modular identities with very large primes are asserted without certificates or code. If the author provides a verifiable certificate or script, I would be inclined to accept; as it stands, the central claim is not independently checkable from the manuscript. I did not find evidence of a mathematical error in the case analysis beyond the presentation issues noted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe main thing to know: this paper does what it claims. It gives explicit M and N such that Mk+N is never a prime plus a Fibonacci number and never a prime plus a Lucas number. That simultaneous version is new, and Remark 1.2 is correct that it is not just the intersection of the existing one-sided arithmetic progressions. The construction is classical Erdős covering-system material, but balancing 33 Fibonacci congruences with 15 Lucas congruences so that the same moduli work for both sequences is real work. The proof after Lemma 2.1 is straightforward, the three F_tables and two L_tables check out, and the contradiction arguments are easy to follow.\n\nThe load-bearing step is Lemma 2.1. Both covering systems must cover every residue class modulo 1152 and 576, respectively. The proof only says 'By calculation, one can easily verify.' No code, no certificate, no enumeration is included. If either system misses one residue, the chain from a representation x=p+F_m or x=q+L_n to the finite prime lists at (3.2)–(3.3) breaks. I looked at the two systems by splitting into 2-adic and 3-adic parts; coverage looks plausible. Direct enumeration is trivial at these moduli, so this is fixable, but the manuscript as submitted does not contain that check. An editor should not have to take the author's word on a computationally decisive step.\n\nMinor issue: the huge moduli are used as primes without proof or certified primality. Standard tests handle them, but it is another unstated computational claim.\n\nNo circularity: the progression comes from CRT on the chosen congruences, and the contradiction tables do not presuppose the conclusion. The citation pattern is fine; [3] supplies inspiration and reference values, and self-citation is justified. The paper is honest about its trial-and-error design in Remark 3.1.\n\nFor whom: researchers in additive number theory working on Romanoff-type problems or covering-system constructions. Not field-shaping, but a solid, explicit contribution. It deserves a serious referee. I would send it out with a request that the author supply an independent verification of Lemma 2.1—a coverage table or a few lines of code. If that lands, the theorem is proved.\n\nI'd probably cite it if my own work touched this exact question.","headline":"A genuine simultaneous Romanoff-type result, explicit and likely correct; the only real gap is an unverified covering-system assertion that a referee can close in an afternoon.","tokens_in":9574,"tokens_out":3589,"would_cite":true,"duration_ms":37282,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11P32","11B39"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single arithmetic progression of positive integers sidesteps both prime-plus-Fibonacci and prime-plus-Lucas representations.","keywords":["Fibonacci numbers","Lucas numbers","arithmetic progressions","covering systems","primes","Romanov-type problems"],"falsifier":"Write a short program that loops over all residues modulo 1152 and checks whether any residue is missed by the 33 congruences in the first system of Lemma 2.1, and likewise loops over residues modulo 576 for the 15-congruence system; a single uncovered residue would invalidate the construction. Independently, verifying each displayed congruence $F_{m_i k+a_i}\\equiv r_i\\pmod{p_i}$ and $L_{n_j k+b_j}\\equiv s_j\\pmod{q_j}$ would confirm the period tables on which the final contradictions rest.","tokens_in":8618,"feed_emoji":"🔢","tokens_out":11576,"duration_ms":106251,"temperature":0.7,"pith_summary":"This paper establishes that one arithmetic progression of positive integers can avoid both additive forms $p+F_m$ and $q+L_n$, where $p,q$ range over primes and $F_m,L_n$ are Fibonacci and Lucas numbers. The paper displays two integers $M$ and $N$ and proves that every number $Mk+N$ with $k\\ge 0$ is expressible neither as a prime plus a Fibonacci number nor as a prime plus a Lucas number. Earlier work gave such arithmetic progressions separately for the Fibonacci case and for the Lucas case, and the paper notes that the combined statement cannot be obtained by intersecting those older progressions. The proof uses covering systems and the Chinese remainder theorem, then rules out all remaining cases through congruences modulo $17$, $19$, and $47$.","feed_headline":"One progression dodges prime-plus-Fibonacci, prime-plus-Lucas sums","feed_subtitle":"Mk+N is never p+F_m or q+L_n for any k, with explicit M and N.","key_machinery":"The machinery is a pair of covering systems combined through the Chinese remainder theorem. A covering system is a finite list of congruences $n\\equiv a_i\\pmod{m_i}$ such that every integer satisfies at least one of them. Lemma 2.1 supplies one system of 33 congruences, with least common multiple 1152, governing Fibonacci indices, and one system of 15 congruences, with least common multiple 576, governing Lucas indices. Since Fibonacci numbers modulo a prime $p_i$ are periodic with period $\\chi_f(p_i)=m_i$, and similarly for Lucas numbers, any representation forces the prime summand to be one of finitely many listed primes. The choice of moduli is balanced so that the largest modulus 1152 is just large enough; the paper notes that it uses all primes whose Fibonacci or Lucas period divides 1152, which is what makes the constants large. The final contradictions come from tables of $F_m$ modulo $17,19,47$ and $L_n$ modulo $19,47$.","core_discovery":"The central discovery is that both representation problems can be disposed of by a single congruence construction. Two covering systems, one of 33 congruences for Fibonacci indices and one of 15 congruences for Lucas indices, force the index of any hypothetical representation into a residue class modulo a listed prime. Because the target number and the forced Fibonacci or Lucas term are congruent modulo that same prime, the prime summand must equal the listed prime itself. The argument then reduces to checking finitely many possibilities of the form $x=p_i+F_{m_i k+a_i}$ or $x=q_j+L_{n_j k+b_j}$, and all of these contradict period tables of the Fibonacci and Lucas sequences modulo $17$, $19$, or $47$. Hence the displayed progression $Mk+N$ lies in the set $B$ of integers with neither representation.","pith_inferences":["The same covering-system pattern may generalize to other linear recurrences or to sums of a prime with two independent fast-growing sequences, provided the moduli are chosen from the periods of the sequences modulo primes.","The paper remarks that the large size of $M$ and $N$ comes from using all primes whose Fibonacci or Lucas period divides 1152; a computer search over alternative covering systems could plausibly produce much smaller progressions.","Since the deciding contradictions use only the small moduli 17, 19, and 47, a natural testable extension is to look for covering systems whose forced primes can be separated by even smaller moduli, which would shrink the constants and might improve quantitative density bounds."],"forward_implications":["The set of positive integers having neither a prime-plus-Fibonacci nor a prime-plus-Lucas representation contains an infinite arithmetic progression.","There is no structural obstruction to merging the Fibonacci and Lucas avoidance problems into one Chinese-remainder construction.","The proof yields separate progressions for the Fibonacci-only and Lucas-only avoiding sets, with much smaller parameters in the Lucas-only case (Corollaries 4.1 through 4.3).","Because the progression is explicit, membership of any particular term can be checked; the paper reports a computer verification of $Mk+N\\in B$ for $0\\le k\\le 10^5$.","Problem 1.4 asks whether the set of integers representable in both ways has positive lower asymptotic density, extending the same quantitative framework."],"supporting_citations":[{"why":"Introduced the covering-system technique that restricts the prime summand to a finite set, the template for this proof.","marker":"[13]"},{"why":"Supplies the covering-system instance and reference values for Fibonacci-index congruences that the author adapts.","marker":"[3]"},{"why":"Proved infinitely many integers are neither $p+f$ nor $-p+f$ for Fibonacci $f$, the Fibonacci-side predecessor.","marker":"[16]"},{"why":"Showed infinitely many integers cannot be expressed as a plus-minus weighted sum of a Fibonacci number and a prime; its ideas inspire the simultaneous treatment.","marker":"[15]"},{"why":"Proved infinitely many integers cannot be written as a plus-minus weighted sum of a Lucas number and a prime, the Lucas-side counterpart.","marker":"[14]"},{"why":"Proved the existence of an arithmetic progression avoiding $p+f$, the closest prior single-progression result now extended by adding Lucas avoidance.","marker":"[20]"}],"fun_headline_variants":["Covering systems forge a progression avoiding all prime-Fibonacci and Lucas sums","One progression avoids every p+F_m and q+L_n representation","Single progression dodges prime-Fibonacci and prime-Lucas sums","Congruence covers yield a progression free of prime-Fibonacci and Lucas sums"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Lemma 2.1's assertion that the two displayed lists of congruences cover all integers modulo 1152 and 576, a check the proof leaves to a computer calculation; if either list misses one residue class, the reduction of every Fibonacci or Lucas index to finitely many primes collapses.","fun_headline_variants_meta":{"raw":{"variants":["Covering systems forge a progression avoiding all prime-Fibonacci and Lucas sums","One progression avoids every p+F_m and q+L_n representation","Single progression dodges prime-Fibonacci and prime-Lucas sums","Congruence covers yield a progression free of prime-Fibonacci and Lucas sums"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00086,"raw_usage":{"total_tokens":3640,"prompt_tokens":762,"completion_tokens":2878,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":378,"completion_tokens_details":{"reasoning_tokens":2799}},"tokens_in":378,"tokens_out":2878,"duration_ms":22465,"temperature":1.0,"reasoning_tokens":2799,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:02:09.740737+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Write a short program that loops over all residues modulo 1152 and checks whether any residue is missed by the 33 congruences in the first system of Lemma 2.1, and likewise loops over residues modulo 576 for the 15-congruence system; a single uncovered residue would invalidate the construction. Independently, verifying each displayed congruence $F_{m_i k+a_i}\\equiv r_i\\pmod{p_i}$ and $L_{n_j k+b_j}\\equiv s_j\\pmod{q_j}$ would confirm the period tables on which the final contradictions rest.","supporting_citations":[{"cited_title":"Erd˝ os, On integers of the form 2 k + p and some related problems, Summa Brasil","cited_arxiv_id":null,"evidence_quote":"Introduced the covering-system technique that restricts the prime summand to a finite set, the template for this proof."},{"cited_title":"Chen, R.-J","cited_arxiv_id":null,"evidence_quote":"Supplies the covering-system instance and reference values for Fibonacci-index congruences that the author adapts."},{"cited_title":"Jones, Fibonacci variations of a conjecture of Polignac, Integers 12 (2012) 659–667","cited_arxiv_id":null,"evidence_quote":"Proved infinitely many integers are neither $p+f$ nor $-p+f$ for Fibonacci $f$, the Fibonacci-side predecessor."},{"cited_title":"Ismailescu, P","cited_arxiv_id":null,"evidence_quote":"Showed infinitely many integers cannot be expressed as a plus-minus weighted sum of a Fibonacci number and a prime; its ideas inspire the simultaneous treatment."},{"cited_title":"Ismailescu, Primefree shifted Lucas sequences of the seco nd kind, J","cited_arxiv_id":null,"evidence_quote":"Proved infinitely many integers cannot be written as a plus-minus weighted sum of a Lucas number and a prime, the Lucas-side counterpart."},{"cited_title":"ˇSiurys, On integers not of the form Fn ± pa, Int","cited_arxiv_id":null,"evidence_quote":"Proved the existence of an arithmetic progression avoiding $p+f$, the closest prior single-progression result now extended by adding Lucas avoidance."}],"review_version":1}