{"id":"891e8dad-6741-4b55-a8f3-54220627624e","arxiv_id":"2411.14308","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For coprime a,b with suitable parity, four terms from the quadratic sequence x(ax+b)/2 or x(ax+b) represent every sufficiently large integer.","lead":"This number theory paper proves that every sufficiently large integer can be written as a sum of four numbers drawn from a wide family of quadratic sequences, an example being w(5w+1)/2. It generalizes Lagrange's four-square theorem, one of the oldest results in additive number theory, with explicit bounds and several concrete corollaries.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.5, Case 2: c=(n-bd)/a is computed as -d0+6q-6r, but the correct term is -6br; for b≡2 (mod 3) the printed choice of r does not force c≡3 (mod 9), so the proof of Theorem 1.5 and its corollaries is incomplete for that residue class.","rationale":"The paper's central theorems 1.1-1.4 appear internally sound; the interval and congruence arguments in Sections 2-3 check out, and the headline corollary based on Theorem 1.1 is not directly affected. The reader's identified gap in Theorem 1.5, Case 2 is the weakest load-bearing point. Independent re-derivation confirms the missing factor b: the printed text uses c=-d0+6q-6r, while substitution gives -6br. For a=3, b=2, n=187353 the printed construction yields c≡6 (mod 9) with 3|d, so Lemma 4.1 cannot be invoked. This is a genuine internal error in a proof step on which Theorem 1.5 and Corollaries 1.9-1.11 depend. It is not a disproof of the theorem: choosing r to solve br≡q+1+d0/3 (mod 3) fixes the congruence and keeps d in K because |K|>18a, so the result is plausibly true with a small correction. The non-reproducibility of finite checks in several corollaries is a separate editorial concern but not the primary mathematical obstruction. The reader's CONDITIONAL verdict is therefore appropriate; no change is recommended.","tokens_in":16263,"tokens_out":21436,"duration_ms":192717,"concrete_test":"Recompute the congruence in Section 4, Case 2 for a=3, b=2, n=187353 using the paper's printed formulas; if the constructed pair has d=585 and c≡6 (mod 9), the proof step fails. Then test the corrected choice r≡2(q+1+d0/3) (mod 3), i.e. r=2, and verify d=603∈K and c≡3 (mod 9); if this succeeds, the gap is localized and repairable, supporting a conditional verdict.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 4, Case 2 (3|a and 3|n), the proof chooses d0∈K with (b-a)d0≡n mod 6a and writes n=(b-a)d0+6aq. Then n-bd0=a(6q-d0). For d=d0+6ar, the correct value is c=(n-bd)/a = (a(6q-d0)-6abr)/a = 6q-d0-6br, not 6q-d0-6r. Lemma 4.1 requires c≡3 (mod 9) because d≡d0≡0 (mod 3). This is equivalent to br≡q+1+d0/3 (mod 3). The paper's choice r≡q+1+d0/3 works only when b≡1 (mod 3); when b≡2 (mod 3), the printed choice does not force c≡3. Concrete instance: a=3, b=2, n=187353 satisfies (1.5) and 3|n. Here K≈(556.13,610.14), d0=567, q=10440, and the paper's r≡1 gives d=585 and c=(187353-2·585)/3=62061≡6 (mod 9). Since d is divisible by 3 but c is not 3 mod 9, this constructed pair fails the hypothesis of Lemma 4.1. The gap affects Theorem 1.5 for all parameters with 3|a, 3|n, b≡2 (mod 3), and hence the corollaries that rely on it. This is a proof gap, not a counterexample: choosing r≡b^{-1}(q+1+d0/3) (mod 3) restores c≡3 (mod 9), and d remains in K because the interval length is >18a.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves several effective asymptotic-basis statements for families of quadratic polynomials. Theorem 1.1 shows that for a>0, b>-a, gcd(a,b)=1, and 2∤ab, the set {x(ax+b)/2 : x∈N} is an asymptotic base of order four, with an explicit bound in (1.1); Theorems 1.2 and 1.3 give analogous statements in the even-a and even-b cases for the unhalved polynomials; Theorem 1.4 handles a coefficient 2 on one variable with gcd(a,5b)=1; and Theorem 1.5 treats a coefficient 3 under a parity hypothesis. The paper derives concrete corollaries, including the headline statement that every integer n>1 is a sum of four generalized second pentagonal numbers, and several exact descriptions of representable sets after finite computer checks. The proofs are elementary, based on variants of Cauchy's four-square lemma and interval arguments.","tokens_in":16715,"tokens_out":12408,"duration_ms":112098,"significance":"If the proofs are completed, the paper makes a meaningful contribution to additive number theory: it provides uniform effective four-square-type theorems for a two-parameter family of quadratic polynomials, extending classical results of Cauchy and Legendre as well as earlier work of Meng-Sun and Krachun-Sun. The bounds are explicit, the finite verifications in the corollaries are checkable, and Theorems 1.1-1.4 appear internally consistent. The principal weakness is the gap in Theorem 1.5, Case 2, which is load-bearing for that theorem and for Corollary 1.9. The gap is localized and appears repairable, so the central thesis of the paper remains plausible, but the manuscript as written is not fully correct.","major_comments":[{"comment":"The displayed computation of c is algebraically wrong for general b. From n=(b-a)d0+6aq and d=d0+6ar, the correct value is c=(n-bd)/a = 6q - d0 - 6br, not 6q - d0 - 6r. Therefore Lemma 4.1's requirement c≡3 (mod 9) is equivalent to br ≡ q + 1 + d0/3 (mod 3). The paper's choice r ≡ q + 1 + d0/3 (mod 3) works only when b≡1 (mod 3). For b≡2 (mod 3) the construction can produce a pair (c,d) with d≡0 (mod 3) but c not ≡ 3 (mod 9); a concrete instance is a=3, b=2, n=187353, where the construction gives d0=567, q=10440, r=1, d=585, and c=(187353-2·585)/3=62061≡6 (mod 9). Since this step is used for all parameters with 3|a, 3|n, and b≡2 (mod 3), Theorem 1.5 and Corollary 1.9 are not proved as written. The gap is local: choosing r with br≡q+1+d0/3 (mod 3) would restore the congruence, and d0+6ar remains in K because the interval length is >18a.","section":"Section 4, Case 2 (proof of Theorem 1.5, p. 16)"}],"minor_comments":[{"comment":"The hypothesis 'with ab or n even' should read 'with ab even or n even'; as printed the parity condition is ungrammatical and could be misread.","section":"Theorem 1.5 statement"},{"comment":"The statement 'any integer /greaterorequalslant...' appears to have lost its variable; the bound should be explicitly attached to n.","section":"Corollary 1.6"},{"comment":"Several corollaries rely on finite computer verifications described only as 'easy computations via a computer'; for reproducibility, the author should specify the verification range and, ideally, provide the checking code or the full list of small exceptional values.","section":"Corollaries 1.1, 1.3, 1.5, 1.10, 1.11"},{"comment":"The arXiv text contains numerous typographical artifacts, including 'RESUL TS' in the header, 'an y' in the abstract, and converted LaTeX symbols such as '/greaterorequalslant' and 'sufficiently'; these should be cleaned in the final version.","section":"Global presentation"},{"comment":"These theorems are asserted without proof, with the note that they follow from Lemma 5.1 and arguments similar to Section 2; since they are not used elsewhere, this is not blocking, but the author should either provide the proofs or clearly label the results as sketched.","section":"Section 5, Theorems 5.1 and 5.2"}],"recommendation":"major_revision","confidential_remarks":"The proof relies on lemmas from two earlier papers coauthored by the author ([2] and [4]); both are published, so this is not circular, but the editor may wish to ensure that those lemmas are independently verified. The gap in Theorem 1.5 is real but appears fixable by the congruence adjustment described in the major comment, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: Theorems 1.1–1.4 look right to me, and the uniform treatment of the two-parameter family x(ax+b)/2 and x(ax+b) is genuinely new. Theorem 1.5 has a concrete algebra slip in Case 2, but it is a fixable proof gap, not a counterexample.\n\nWhat the paper actually does: for coprime a,b with parity conditions, it shows the fourfold sums of the quadratic sequence form an asymptotic basis, with explicit thresholds. The interval arguments in Section 2 are detailed and check out; the extension of Krachun–Sun to general a,b in Theorem 1.4 is also plausible. The corollaries with explicit exceptional sets are nice, though the finite verifications are asserted without code or data.\n\nThe soft spot is real. In Section 4, Case 2 (3|a, 3|n), the paper writes c=(n−bd)/a = −d0 + 6q − 6r, but the last term should be −6br. With b≡2 (mod 3), the chosen r does not force c≡3 (mod 9), so Lemma 4.1's condition can fail. The stress-test instance a=3, b=2, n=187353 gives d=585 and c≡6 (mod 9), which confirms the failure. This affects Theorem 1.5 for all parameters in that residue class and hence the general Corollary 1.9. It is easily repaired: choose r ≡ b^{-1}(q+1+d0/3) (mod 3) instead, and the same interval-length argument keeps d in K. I'd ask the author to fix that and to double-check the congruence display.\n\nThe self-citation load is heavier than I'd like—three lemmas come from the author's earlier papers—but those are published and the lemmas are used as black boxes, which is normal for this area. The main theorems do not assume their own conclusion.\n\nRecommendation: send it to review. The gap is localized and fixable, the central results are worth having, and the explicit bounds are a useful addition. For the finite corpus checks, ask for machine-readable verification or at least a reproducible script.","headline":"Broadens the Lagrange-style four-square family to a two-parameter quadratic sequence, and the central theorems hold up; Theorem 1.5 has a localized but fixable congruence slip.","tokens_in":17255,"tokens_out":6021,"would_cite":true,"duration_ms":52278,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B13","11E25","11D85","11E20","11P05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that every sufficiently large integer is a sum of four values of a single quadratic polynomial from a broad family, generalizing the four-square theorem.","keywords":["additive basis","asymptotic basis","polygonal numbers","second pentagonal numbers","quadratic polynomials","four-square theorem","sums of four values","number theory"],"falsifier":"For $a=3$, $b=2$, and $n=187353$, the construction in the proof yields $d=585$ and $c\\equiv6\\pmod9$, so the required $c\\equiv3\\pmod9$ fails; checking whether this $n$ is actually representable as $3w(3w+2)+x(3x+2)+y(3y+2)+z(3z+2)$ would show whether Theorem 1.5 is false or merely unproved for $b\\equiv2\\pmod3$.","tokens_in":16048,"feed_emoji":"🔢","tokens_out":11705,"duration_ms":104011,"temperature":0.7,"pith_summary":"This paper establishes new four-term additive results for families of quadratic numbers, in the spirit of the classical theorem that every nonnegative integer is a sum of four squares. The main theorem says that for coprime parameters $a,b$ with $ab$ odd, the set of numbers $x(ax+b)/2$ with $x$ nonnegative is an asymptotic basis of order four, with an explicit threshold; the analogue for $x(ax+b)$ holds when $a$ is even and $b$ is odd. The headline instance is the second pentagonal numbers $w(5w+1)/2$: every integer $n>1$ is a sum of four such generalized numbers with integer coefficients. The proof selects an auxiliary integer $d$ and, by a congruence condition, finds $c$ so that $n=ac+bd$, then invokes a four-squares-with-fixed-sum lemma to complete the representation.","feed_headline":"Every integer above 1 is a sum of four second pentagonal numbers","feed_subtitle":"A four-square-style theorem for a broad family of quadratic values, proved with explicit thresholds.","key_machinery":"The machinery is a reduction to a four-square-with-prescribed-sum lemma. If $c$ and $d$ have the same parity, $4\\nmid c$, and $d^2/4<c<(d^2+2d+4)/3$, then $c$ is a sum of four squares $w^2+x^2+y^2+z^2$ and $d$ is the sum $w+x+y+z$. The paper writes the target $n$ as $ac+bd$, chooses $d$ in a carefully positioned interval whose length exceeds the modulus step, and solves a congruence modulo $2a$ (or $4a$ or $6a$) so the resulting $c,d$ satisfy the lemma. The identity $n=a(w^2+x^2+y^2+z^2)+b(w+x+y+z)$ then becomes the desired sum $w(aw+b)/2+x(ax+b)/2+y(ay+b)/2+z(az+b)/2$. The weighted variants use analogous interval and congruence arguments with $5$ and $9$ divisors.","core_discovery":"The central claim is that a single quadratic family $\\{x(ax+b)/2 : x\\in\\mathbb{N}\\}$, for coprime $a,b$ with the right parity, represents all sufficiently large integers as a sum of four elements. The flagship case is the second pentagonal numbers: with $(a,b)=(5,1)$, every integer $n>1$ can be written as $w(5w+1)/2+x(5x+1)/2+y(5y+1)/2+z(5z+1)/2$ with $w,x,y,z\\in\\mathbb{Z}$. The same method yields weighted variants, including sums where one summand carries a factor $2$ or where the first coefficient is $3$, each with explicit thresholds and short exception lists. The theorems are effective: they give bounds rather than only a statement of eventual representability.","pith_inferences":["The explicit bounds are likely far from sharp: the interval-length method typically pays large constants, and the paper's own numerical corollaries suggest the true thresholds are much smaller for small $a,b$.","The congruence difficulty in the coefficient-$3$ theorem appears repairable by a different residue choice; if repaired, the coefficient-$3$ statement would likely hold for all odd coprime pairs, matching the exception lists the paper derives.","The same reduction template could apply to any quadratic whose values pair with four-square-sum data through $n=ac+bd$, suggesting that the class of four-term asymptotic bases of order four may be much broader than the families listed in the theorems."],"forward_implications":["For $(a,b)=(5,1)$, every integer $n>1$ is a sum of four generalized second pentagonal numbers, settling the second-pentagonal analogue of the four-square theorem.","For each admissible $(a,b)$, the explicit threshold in Theorem 1.1 turns the asymptotic statement into a finite verification problem; the corollaries show such verifications produce clean exception lists such as $\\{1,3,5,10,12,20,25,27,38,53,65,153,165\\}$.","The weighted theorems show that replacing one coefficient by $2$ or $3$ preserves four-term universality over the integers for all large targets, matching the earlier coefficient-$1$ results.","In the even-$b$ case, Theorem 1.3 implies the same family is an asymptotic base of order five, because one of $n$ or $n-(a+b)$ is never divisible by $4$."],"supporting_citations":[{"why":"Supplies the four-square-with-fixed-sum lemmas at the core of Theorems 1.1, 1.3, and 1.5.","marker":"[4]"},{"why":"Gives the classical polygonal-number theorem and the four-square-sum lemma it extends.","marker":"[6]"},{"why":"Supplies the even-number lemma used in Theorem 1.4 and the method that theorem adapts.","marker":"[2]"},{"why":"Provides the exact formulas for four polygonal-number representations that frame the modern setting.","marker":"[3]"},{"why":"Proved the analogous four-term result for octagonal numbers that motivates this family of statements.","marker":"[8]"}],"fun_headline_variants":["Four pentagonal numbers sum to any integer above 1","Lagrange-style theorem for generalized pentagonal numbers","Every large integer is four generalized pentagonal numbers","A four-square analog for quadratic sequences"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the coefficient-three theorem assumes a divisibility identity that only holds for $b=1$; for many allowed values of $b$ the constructed numbers violate the condition the final lemma needs, so the claim is not established for those parameters.","fun_headline_variants_meta":{"raw":{"variants":["Four pentagonal numbers sum to any integer above 1","Lagrange-style theorem for generalized pentagonal numbers","Every large integer is four generalized pentagonal numbers","A four-square analog for quadratic sequences"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000174,"raw_usage":{"total_tokens":1263,"prompt_tokens":907,"completion_tokens":356,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":523,"completion_tokens_details":{"reasoning_tokens":297}},"tokens_in":523,"tokens_out":356,"duration_ms":3387,"temperature":1.0,"reasoning_tokens":297,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:21:41.017455+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $a=3$, $b=2$, and $n=187353$, the construction in the proof yields $d=585$ and $c\\equiv6\\pmod9$, so the required $c\\equiv3\\pmod9$ fails; checking whether this $n$ is actually representable as $3w(3w+2)+x(3x+2)+y(3y+2)+z(3z+2)$ would show whether Theorem 1.5 is false or merely unproved for $b\\equiv2\\pmod3$.","supporting_citations":[{"cited_title":"Meng and Z.-W","cited_arxiv_id":null,"evidence_quote":"Supplies the four-square-with-fixed-sum lemmas at the core of Theorems 1.1, 1.3, and 1.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the classical polygonal-number theorem and the four-square-sum lemma it extends."},{"cited_title":"Krachun and Z.-W","cited_arxiv_id":null,"evidence_quote":"Supplies the even-number lemma used in Theorem 1.4 and the method that theorem adapts."},{"cited_title":"Sums of four polygonal numbers: precise formulas","cited_arxiv_id":"2405.14710","evidence_quote":"Provides the exact formulas for four polygonal-number representations that frame the modern setting."},{"cited_title":"Sun, A result similar to Lagrange’s theorem , J","cited_arxiv_id":null,"evidence_quote":"Proved the analogous four-term result for octagonal numbers that motivates this family of statements."}],"review_version":1}