{"id":"2ecbe6cf-2ecb-4239-b997-72c31d7f6c41","arxiv_id":"2506.03631","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For n = p + f_{k1^2} + f_{k2^2}, the paper shows infinitely many n have zero representations and positive asymptotic densities have one or at least two representations.","lead":"This paper proves that some integers can never be written as a prime plus two Fibonacci numbers whose position numbers are perfect squares, while a positive fraction can be written in exactly one way and another positive fraction in at least two ways. It extends classical Romanoff-type results to a new class of additive representations using congruence covering arguments and sieve estimates.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The uniqueness assertion in Lemma 3.7 is load-bearing but not proven: the one-line recurrence justification does not rule out Fibonacci-sum collisions, and the entire second-moment bound depends on excluding them.","rationale":"The reader's weakest-assumption identification matches the main risk I see: Lemma 3.7's uniqueness claim is not established by the given sentence, and it is genuinely load-bearing for the second-moment estimate and hence for Theorem 1.2. The claim is likely true, and the K_i restrictions are probably sufficient, but the manuscript does not show this. I am not moving the verdict because the proof structure is otherwise plausible, the constants in Lemma 3.6 check out at the level of the displayed estimates, and the remaining large finite computations in Theorem 2.1, while unverified, are routine in this area. A conditional verdict with a request to supply the missing uniqueness proof is appropriate; if a counterexample to the uniqueness claim were found, the paper would need substantial revision.","tokens_in":15372,"tokens_out":32618,"duration_ms":337454,"concrete_test":"Independently derive the uniqueness claim in Lemma 3.7: classify all integer solutions of F_x+F_y=F_z+F_w and show the only nontrivial families are of the form F_{n-1}+F_{n+2}=2F_{n+1} (up to symmetry), then verify that no such family can have one index in K1^2 and one in K2^2, since K1^2 and K2^2 are disjoint and differ modulo 3 and 9. Cross-check this analytically derived conclusion with a brute-force search over all allowed k<=10^5; if any collision with four indices in the permitted classes appears, Lemma 3.7 is false as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 3.7 begins by asserting that f_{k1^2}+f_{k2^2}=f_{l1^2}+f_{l2^2} with k_i,l_i in K_i forces k_i=l_i, justified only by \"by f_m=f_{m-1}+f_{m-2} for all m>=2.\" This assertion is load-bearing: it is what allows the proof to replace r(n)(r(n)-1) by a sum over p1 neq p2. If it fails, then pairs with p1=p2 and distinct Fibonacci-index pairs contribute to r(n)(r(n)-1) but are omitted, so the bound 0.2322 pi^2 / (...) x in Lemma 3.7, and hence the positive density in Theorem 1.2, would not follow. The cited recurrence is not a proof of injectivity: for example, the recurrence itself gives F_{n-1}+F_{n+2}=2F_{n+1}, a nontrivial two-term Fibonacci-sum equality. The restrictions k_i in K_i may well rule out all such collisions, but the paper does not supply the needed argument, e.g., a classification of solutions to F_x+F_y=F_z+F_w or a valuation/residue proof. I am not claiming the assertion is false; I am claiming it is essential, nontrivial, and currently unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the representation function r(n) counting representations of a positive integer n as p + f_{k_1^2} + f_{k_2^2}, where p is prime and k_1, k_2 are nonnegative integers with k_1 ≤ k_2. The main results are: Theorem 1.1 gives an infinite arithmetic progression of n with r(n)=0; Theorem 1.2 shows that a positive proportion of integers are represented uniquely; Theorem 1.3 shows that a positive proportion are represented in at least two ways. The proofs are built on a finite congruence covering (Theorem 2.1) that characterizes the allowed indices and primes for n ≡ n_0 (mod M), followed by first and second moment estimates using the prime number theorem in arithmetic progressions and sieve bounds.","tokens_in":15681,"tokens_out":19273,"duration_ms":192566,"significance":"If the proofs are correct, the results are a noteworthy analogue of classical Romanoff-type theorems: despite the extreme sparsity of the sequence {F_{n^2}}, it still behaves like an additive basis in a density sense. The paper is clearly organized, supplies explicit constants and moduli, and the main technical achievement is a second-moment estimate that is both elaborate and mostly explicit. The numerical bounds, such as Lemma 3.6, are derived in detail and appear plausible. The overall strategy is sound, provided the unresolved uniqueness issue in Lemma 3.7 and the related issue in Lemma 4.2 are fixed.","major_comments":[{"comment":"The proof of Lemma 3.7 opens with the assertion that if f_{k_1^2}+f_{k_2^2}=f_{l_1^2}+f_{l_2^2} with k_i,l_i ∈ K_i, then k_i = l_i, justified only by 'by f_m=f_{m-1}+f_{m-2} for all m≥2'. This is not a proof: the Fibonacci recurrence alone permits non-trivial equalities such as F_{n-1}+F_{n+2}=F_{n+1}+F_{n+1} and F_{n+1}+F_{n+2}=F_0+F_{n+3}. The restrictions that the indices are squares in the residue classes K_i may well rule out such collisions, but the paper supplies no argument to that effect. This assertion is load-bearing: it is exactly what allows the reduction of r(n)(r(n)-1) to the case p_1 ≠ p_2. If the assertion fails, the bound in Lemma 3.7, and consequently the positive density in Theorem 1.2, would not follow. A rigorous proof is needed, for instance a classification of all solutions to F_x+F_y=F_z+F_w with x,y,z,w in the relevant sets, or a valuation/residue argument exploiting that the indices are squares.","section":"Section 3, Lemma 3.7"},{"comment":"In the proof of Lemma 4.2, the contribution to ∑_{n≤x} r(n)^2 from tuples with p_1 = p_2 is bounded by O(x) without justification. This term counts, for each prime p, pairs of index pairs (k_1,k_2) and (l_1,l_2) with f_{k_1^2}+f_{k_2^2}=f_{l_1^2}+f_{l_2^2}. Since the number of square-index pairs with sum ≤ x is O(log x), an a priori bound of O(x log x) is trivial; obtaining O(x) requires an upper bound on the number of representations of an integer as a sum of two Fibonacci numbers with square indices. No such bound is proved or cited. This is essentially the same uniqueness/collision question as in Lemma 3.7, and it is load-bearing for the estimate ∑ r(n)^2 ≪ x needed in the proof of Theorem 1.3. The authors should supply an argument that the number of non-diagonal collisions is O(x), or otherwise adjust the proof.","section":"Section 4, Lemma 4.2"}],"minor_comments":[{"comment":"The statement u(769)=192 is used without proof or verification; please add a short justification (e.g., a computation of F_{192} mod 769 and F_{193} mod 769) or a citation.","section":"Section 2, Proof of Theorem 1.1"},{"comment":"The expression '5/2 log log(f_1...f_n)' is ambiguous; it should be written as 5/(2 log log(f_1...f_n)) to match the bound from [17, (3.41)].","section":"Section 3, Lemma 3.6"},{"comment":"The abstract says 'positive asymptotic densities', but the proofs establish positive lower density (positive proportion); consider rephrasing to 'positive lower density' or 'positive proportion' for precision.","section":"Abstract"},{"comment":"The notation P(d) for the largest prime divisor of d is used before it is defined; please define it at the point of first use.","section":"Section 3, Lemma 3.7"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal. The central issue is the unproven uniqueness assertion in Lemma 3.7 and its counterpart in Lemma 4.2 regarding solutions of f_{x}+f_{y}=f_{z}+f_{w} with square indices in the specified residue classes. If the authors can supply a rigorous proof of the needed uniqueness or collision bound, the paper would be a solid contribution. I would also appreciate the editor's attention to the verification of the purely numerical values (u(769)=192 and the bound in Lemma 3.6), which appear plausible but are not fully documented."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline: this is a genuine new result in the Romanoff-type line, with a real gap in the proof of the second moment bound. The main construction—Theorem 2.1's congruence covering—is careful and explicit, and Theorems 1.1–1.3 would follow cleanly if Lemma 3.7 held as stated.\n\nWhat is new: no one has treated p plus two Fibonacci numbers with square indices. The prior work on p+F_k and p+2^{a^2}+2^{b^2} does not make this routine; the rank-of-apparition covering needs new residue computations, and the authors supply them. The density results for r(n)=1 and r(n)≥2 are natural analogues of the Chen–Xu conjecture and worth having.\n\nThe soft spot is exactly what the stress-test flags. In Lemma 3.7, the assertion that f_{k_1^2}+f_{k_2^2}=f_{l_1^2}+f_{l_2^2} with k_i,l_i in K_i forces k_i=l_i is justified by one sentence citing the recurrence f_m=f_{m-1}+f_{m-2}. That is not a proof. The recurrence alone permits two-term sum collisions (e.g., F_n+F_{n+3}=F_{n+2}+F_{n+2}), and nothing in the text rules out such identities when the indices are squares in those congruence classes. This assertion is load-bearing: it is what lets the proof drop the p1=p2 terms in r(n)(r(n)-1). If the assertion fails, the second moment gets an additional term that is not controlled, and the positive density in Theorem 1.2 no longer follows from the written argument.\n\nI don't think the assertion is obviously false; the square-index restriction plus the congruence classes may well eliminate all collisions. But the authors need to prove it, e.g., by citing a classification of F_x+F_y=F_z+F_w or by a valuation argument. As written, the gap is nontrivial and central.\n\nMinor point: the large finite residue computations in Theorem 2.1 are stated but not carried out. That is acceptable at this level because they are explicit and checkable, but a referee should ask for a verification script or a reproducibility note.\n\nWho this is for: people working on Romanoff-type problems and additive bases with sparse sequences. The paper deserves a serious referee; the gap is likely repairable, but the referee should push on Lemma 3.7 before acceptance.\n\nRecommendation: send to peer review, and make the handling editor aware that the proof of Lemma 3.7 needs a substantive addition.","headline":"Genuine new result for sums of a prime and two Fibonacci numbers with square indices, but the second moment bound in Lemma 3.7 rests on an unproved uniqueness assertion that needs a real proof.","tokens_in":16195,"tokens_out":8191,"would_cite":false,"duration_ms":78832,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11P32","11A41","11B39","11B13"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves three simultaneous facts about the representation function $r(n)$: an infinite arithmetic progression with no representations, a positive proportion with exactly one representation, and a positive proportion with at least…","keywords":["Romanoff type problems","primes","Fibonacci numbers","applications of sieve methods","asymptotic density","representation functions","Fibonacci squares","additive bases"],"falsifier":"Take the unique residue $n_1\\pmod{769M}$ with $n_1\\equiv n_0\\pmod M$ and $n_1\\equiv501\\pmod{769}$, and compute $r(n_1+769Mk)$ for the first few $k$; a single value above zero would disprove Theorem 1.1. A separate check is to enumerate $k_i,\\ell_i\\le10^6$ in the two index classes and test whether $f_{k_1^2}+f_{k_2^2}=f_{\\ell_1^2}+f_{\\ell_2^2}$ occurs with distinct pairs, which would invalidate the collision assertion used in Lemma 3.7.","tokens_in":15202,"feed_emoji":"🔢","tokens_out":16495,"duration_ms":158099,"temperature":0.7,"pith_summary":"This paper studies which integers can be written as a prime plus two Fibonacci numbers whose indices are perfect squares, $n=p+f_{k_1^2}+f_{k_2^2}$, and counts representations with $r(n)$. It proves that an entire arithmetic progression of positive integers has $r(n)=0$, that a positive proportion of integers have $r(n)=1$, and that a positive proportion have $r(n)\\ge 2$. The interest is that the Fibonacci-square subsequence is very sparse, yet together with the primes it still covers a positive proportion of integers in a way that is both unique and multiple on positive-density sets. The proofs combine a congruence obstruction built from Fibonacci periodicity with sieve estimates that control collisions between different representations.","feed_headline":"Primes plus two Fibonacci squares: an infinite gap, positive density","feed_subtitle":"A congruence obstruction creates an unrepresentable arithmetic progression, yet unique and repeated representations each occur with…","key_machinery":"The central object is a large modulus $M=621386267972593776074029725204132260351094$, the product of sixteen primes, together with a residue $n_0\\pmod M$ and a prime residue $p_0\\pmod M$ determined by the Chinese remainder theorem. The mechanism is Fibonacci periodicity: for each prime $q\\mid M$, the Fibonacci sequence modulo $q$ has period $u(q)$, the least positive $u$ with $f_u\\equiv f_0$ and $f_{u+1}\\equiv f_1\\pmod q$, and $v(d)=\\max_{p\\mid d}u(p)$ measures how far a modulus $d$ can detect a difference $f_{k_1^2}-f_{h_1^2}$. The proof forces $n\\equiv n_0\\pmod M$ to imply $k_i\\equiv0,64,128\\pmod{192}$ and $p\\equiv p_0\\pmod M$, then uses an upper-bound sieve for prime pairs to show that collisions $f_{k_1^2}+f_{k_2^2}=f_{\\ell_1^2}+f_{\\ell_2^2}$ are rare. The numerical estimate $\\sum_{d\\ge1}\\mu^2(d)/(d\\sqrt{[192^2,v(d)]})<0.23219$, where $[\\cdot,\\cdot]$ denotes the least common multiple, is what makes the second moment smaller than the first, producing positive density.","core_discovery":"On the paper's own terms, the discovery is that the sparse set $\\{f_{k^2}:k\\in\\mathbb N_0\\}$ behaves like a two-term additive basis for the primes in a strong density sense. The paper constructs a modulus $M$ and a residue class $n_0\\pmod M$ with the property that any representation of such an $n$ forces the prime to lie in a single residue class and both square indices to lie in two prescribed residue classes modulo $192$; intersecting that class with a further modulus $769$ produces an arithmetic progression with no representations at all. Averaging $r(n)$ over the residue class gives a positive first moment asymptotic to a constant times $x$, while a sieve bound on coincidences gives a second moment of order $x$; because the second-moment constant is small enough, both $\\{n:r(n)=1\\}$ and $\\{n:r(n)\\ge2\\}$ acquire positive asymptotic density within the class, hence in the integers.","pith_inferences":["The same CRT-plus-periodicity construction should transfer to $p+f_{k_1^s}+f_{k_2^s}$ for other exponents $s$: one needs a modulus whose Fibonacci periods divide a common multiple of the residues of $k^s$, and the index classes would force the same obstruction; the square case is one instance of a general template.","The numerical threshold in Lemma 3.6 is compared with $9\\log\\alpha$ at the end of the proof of Theorem 1.2, so the lower density of unique representations could be increased by sharpening either the sieve constant in Lemma 3.5 or the divisor-sum estimate; the paper's constants are not optimal.","Problem 1.5, asking whether every multiplicity $m$ occurs with positive proportion, remains open for $m\\ge3$; the present proof only controls the second moment, so higher multiplicities would require higher moments or a different argument.","The method's dependence on Fibonacci periodicity suggests the same density phenomenon should hold for other Lucas sequences; the decisive quantity is whether the associated $v(d)$ satisfies a divisor-sum bound like Lemma 3.6."],"forward_implications":["An infinite arithmetic progression of positive integers is missed entirely, so $\\{p+f_{k_1^2}+f_{k_2^2}\\}$ is not an additive basis in the strong sense of eventually representing all integers.","A positive proportion of all positive integers are represented exactly once, and a positive proportion are represented at least twice; both layers of the representation function are visible at the level of asymptotic density.","For the residue class $n\\equiv n_0\\pmod M$ the average count is asymptotic to $x/(2^{11}3^2\\varphi(M)\\log\\alpha)$, and the second-moment bound implies that the set with $r(n)\\ge T$ has density $O(1/T^2)$.","The boundary case of two square Fibonacci indices, where the reciprocals of the exponents sum to $1$, now has the finer description of unique and multiple representations alongside a missing progression, while the earlier result only gave positive proportion of represented integers."],"supporting_citations":[{"why":"Supplies the original setting of a prime plus a power of two with positive-density representations, which the present counting function generalizes.","marker":"[16]"},{"why":"Provides the model for producing an infinite arithmetic progression of integers with no representation in a prime-plus-power-of-two family.","marker":"[10]"},{"why":"Previous result on positive proportions of integers represented as a prime plus a sum of Fibonacci powers, the direct antecedent for the two-square case.","marker":"[5]"},{"why":"Theorem 3.12 in this monograph gives the uniform upper-bound sieve for primes $p$ with $p+m$ prime, used in Lemma 3.4 and Lemma 3.5.","marker":"[11]"},{"why":"Proposition 2.1 bounds the number of solutions to $y^2\\equiv a\\pmod K$, used as Lemma 3.3 to count admissible $k_1$.","marker":"[8]"},{"why":"Provides the bound $|B_d|\\le4$ on Fibonacci residues modulo a prime, used in Lemma 3.7 to control $k_1^2$ modulo $v(d)$.","marker":"[18]"},{"why":"Alternative source for the same $|B_d|\\le4$ bound on Fibonacci residues modulo a prime.","marker":"[20]"},{"why":"Supplies the inequality for $k/\\varphi(k)$ used in the numerical estimate in Lemma 3.6.","marker":"[17]"},{"why":"Supplies the prime number theorem for arithmetic progressions used in Lemma 3.2 to evaluate the average of $r(n)$.","marker":"[1]"}],"fun_headline_variants":["Prime plus two Fibonacci squares: infinite gap, positive densities","Prime sums with Fibonacci squares: infinite missing set, positive densities","Infinite unrepresentable progression, positive densities for prime + 2 Fibonacci squares","Fibonacci-square prime sums: a void progression and dense representations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument needs the fact that two different pairs of square-indexed Fibonacci numbers, with indices chosen from the special allowed sets, never add to the same total; the paper supports this only by citing the recurrence $f_m=f_{m-1}+f_{m-2}$, which by itself allows collisions such as $F_n+F_{n-3}=2F_{n-1}$.","fun_headline_variants_meta":{"raw":{"variants":["Prime plus two Fibonacci squares: infinite gap, positive densities","Prime sums with Fibonacci squares: infinite missing set, positive densities","Infinite unrepresentable progression, positive densities for prime + 2 Fibonacci squares","Fibonacci-square prime sums: a void progression and dense representations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00238,"raw_usage":{"total_tokens":9115,"prompt_tokens":856,"completion_tokens":8259,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":472,"completion_tokens_details":{"reasoning_tokens":8186}},"tokens_in":472,"tokens_out":8259,"duration_ms":61100,"temperature":1.0,"reasoning_tokens":8186,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:00:41.755646+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the unique residue $n_1\\pmod{769M}$ with $n_1\\equiv n_0\\pmod M$ and $n_1\\equiv501\\pmod{769}$, and compute $r(n_1+769Mk)$ for the first few $k$; a single value above zero would disprove Theorem 1.1. A separate check is to enumerate $k_i,\\ell_i\\le10^6$ in the two index classes and test whether $f_{k_1^2}+f_{k_2^2}=f_{\\ell_1^2}+f_{\\ell_2^2}$ occurs with distinct pairs, which would invalidate the collision assertion used in Lemma 3.7.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the original setting of a prime plus a power of two with positive-density representations, which the present counting function generalizes."},{"cited_title":"Erd˝ os,On integers of the form2 k +pand some related problems, Summa Brasil","cited_arxiv_id":null,"evidence_quote":"Provides the model for producing an infinite arithmetic progression of integers with no representation in a prime-plus-power-of-two family."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Previous result on positive proportions of integers represented as a prime plus a sum of Fibonacci powers, the direct antecedent for the two-square case."},{"cited_title":"Halberstam and H","cited_arxiv_id":null,"evidence_quote":"Theorem 3.12 in this monograph gives the uniform upper-bound sieve for primes $p$ with $p+m$ prime, used in Lemma 3.4 and Lemma 3.5."},{"cited_title":"Ding,On a problem of Romanoff type, Acta Arith","cited_arxiv_id":null,"evidence_quote":"Proposition 2.1 bounds the number of solutions to $y^2\\equiv a\\pmod K$, used as Lemma 3.3 to count admissible $k_1$."},{"cited_title":"Schinzel,Special Lucas sequences, including the Fibonacci sequence, modulo a prime, inA Tribute to Paul Erd˝ os, eds","cited_arxiv_id":null,"evidence_quote":"Provides the bound $|B_d|\\le4$ on Fibonacci residues modulo a prime, used in Lemma 3.7 to control $k_1^2$ modulo $v(d)$."},{"cited_title":"Somer,Distribution of residues of certain second-order linear re- currences modulop, inApplications of Fibonacci Numbers, eds","cited_arxiv_id":null,"evidence_quote":"Alternative source for the same $|B_d|\\le4$ bound on Fibonacci residues modulo a prime."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the inequality for $k/\\varphi(k)$ used in the numerical estimate in Lemma 3.6."},{"cited_title":"Apostol, Introduction to analytic number theory, Springer- Verlag, 1976","cited_arxiv_id":null,"evidence_quote":"Supplies the prime number theorem for arithmetic progressions used in Lemma 3.2 to evaluate the average of $r(n)$."}],"review_version":1}