{"id":"588128f9-c39f-4465-8088-307ee8ae9205","arxiv_id":"2607.29566","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"For an integer with k odd prime factors, the least quadratic residue lies between 4^{k-o(k)} and O(k^2 4^k), giving new bounds for representing integers by binary quadratic forms.","lead":"This paper proves near-sharp bounds for the least quadratic residue modulo a composite number, showing it is essentially determined by 4^k where k is the number of prime factors. The result also yields limits on how small a discriminant a binary quadratic form can have while still representing every integer up to N.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unconditional Theorem 1.2 rests on the imported log-free zero-density estimate (9); with that premise the partition argument is sound, so no internal flaw found.","rationale":"The reader's weakest assumption is the right one. I checked the main line of proof: the pigeonhole upper bound, the finite-field construction in Proposition 2.1 (including Lemma 2.2), the GRH-conditional argument, and the quadratic-form applications in Section 4 all hold together. The only place where the central claim imports an unproved deep input is (9). Since (9) is a standard theorem in the field and the cited reference is apt, the risk is acceptable for an ACCEPT verdict; no change to the reader's verdict is needed.","tokens_in":10475,"tokens_out":39020,"duration_ms":376689,"concrete_test":"Take the specific rectangle families used in the proof (σ≥1-j/logΔ, |t|≤Δ for j=1..J), derive the number of fundamental discriminants |D|≤Δ with a zero in each shell directly from Gallagher's density theorem, and verify (9) holds with constants C1,C2 independent of j and no extra logΔ power. Then recompute the sum over j of k_j log x_j; if it still fits C logΔ(log logΔ)^2, the unconditional bound stands, and otherwise the proof needs a stronger zero-density input.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The unconditional half of Theorem 1.2 is exactly as strong as the exterior estimate (9): it partitions all fundamental discriminants |D|≤Δ into shells D_j and needs |D_j|≤C1 exp(C2 e_j). This bound controls the number of prime factors k_j needed in Lemma 3.1; summing k_j log x_j over j yields the claimed N(Δ)=exp(C logΔ (log logΔ)^2). If the true density of discriminants in a shell were worse than exponential in e_j, or if Gallagher's estimate had to be applied with an extra logΔ factor, the constructed modulus would no longer be below the claimed bound. The estimate is standard and the citation is appropriate, so risk is low; however it is the one place where the central unconditional claim imports a deep theorem rather than proving it. The rest of the argument, including Proposition 2.1/Lemma 2.2 for the lower bound and the GRH conditional argument, is internally coherent.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies ℓ(n), the least squarefree non-trivial quadratic residue modulo n. Theorem 1.1 gives the upper bound ℓ(n) ≤ C k^2 4^k, where k is the number of odd prime factors of n, and constructs infinitely many n with exactly k odd prime factors for which ℓ(n) ≥ 4^{k − Ck/log k}. Theorem 1.2 constructs, for a given Δ, an odd squarefree modulus n of size at most exp(C logΔ (loglogΔ)^2) such that every fundamental discriminant D with 1 < |D| ≤ Δ has χ_D(p) = −1 for some prime p | n; under GRH the bound improves to exp(C logΔ loglogΔ) for every C > 2/log2. This yields lower bounds for ℓ(n) in terms of n. Theorem 1.3 converts these results into statements about binary quadratic forms: positive results on the discriminant needed to represent every integer up to N, and a negative result for discriminants up to roughly Δ = exp(c log N/(loglog N)^2), with a GRH exponent improvement. The proofs combine a pigeonhole argument, a finite-field construction using irreducible polynomials over F2, a GRH/zero-density shell argument, and classical facts on binary quadratic forms.","tokens_in":10690,"tokens_out":48162,"duration_ms":466558,"significance":"If correct, this is a substantial advance. The lower bound in Theorem 1.1 is striking: although the density of quadratic residues is about 2^{−k}, the authors construct moduli for which ℓ(n) is about 4^{k−o(k)}. The finite-field Lemma 2.2 gives an elegant and nearly sharp mechanism for this. The unconditional Theorem 1.2 is exactly as strong as the imported log-free zero-density estimate (9); with that standard input, the shell argument is internally coherent. The applications in Theorem 1.3 are nontrivial and appear to be correct, with the right constants in the GRH-conditional part. The proofs are self-contained apart from the cited zero-density estimate and standard analytic lemmas; I found no load-bearing mathematical error beyond the missing definition noted below.","major_comments":[{"comment":"The shell parameter e_j appears immediately after 'Put J = ...' in the definition of β_j, and is then used in the density estimate (9), in x_j = Δ^{j/e_{j−1}}, and in the bound (10), but it is never defined in the manuscript as written. From J = ceil(log((1/2)logΔ)) and the factor e·j in (10), I infer that the intended definition is e_j = e^j. With that definition the inequalities in Lemma 3.2 and the final bound log n ≤ C logΔ (loglogΔ)^2 check out; for example, the j = 1 shell works because log 10 < e. However, as printed the proof of the unconditional part of Theorem 1.2 cannot be checked. Please define e_j explicitly and clarify that the 'e·j' in (10) is Euler's number times j.","section":"§3, unconditional part"}],"minor_comments":[{"comment":"The illustrative sentence says that 5, 7, and 11 are associated to 'the three irreducible polynomials of degree 3'. Over F2 there are only two irreducible cubics; the next available irreducibles are the three irreducible quartics. The proof is unaffected, but the example should be corrected.","section":"§2, Proposition 2.1"},{"comment":"The passage from (7) to (8) is compressed: (7) is the Mellin integral, which equals log x times the weighted prime sum, and the final bound for Σ_p (log p/p)χ_D(p)w(...) requires dividing the bound for the zero sum by log x. The final estimate is correct, but this division should be made explicit for readability.","section":"§3, Lemma 3.2"},{"comment":"When deriving the lower bound for ℓ(n), the authors note that either ℓ(n) or 4ℓ(n) is fundamental. If ℓ(n) ≡ 3 mod 4, the fundamental discriminant is 4ℓ(n), and the construction only directly rules out 4ℓ(n) ≤ Δ. This changes Δ by at most a factor of 4 and is absorbed into constants, but the text should state this.","section":"§3, after Theorem 1.2"},{"comment":"The claims L(4) = 570 and L(5) = 2679 are stated without proof or a description of the verification method. Since these values are not needed for the main results, please either provide a short certificate or explicitly label them as computational observations.","section":"§5, Further questions"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Plain verdict: this is a results paper. The finite-field construction in Proposition 2.1 — using irreducible polynomials over F2[x] and Hayes-type characters to make ℓ(n) nearly as large as 4^k — is the genuinely new piece. The pigeonhole upper bound in Theorem 1.1 is elementary and correctly handled. The transfer to binary quadratic forms in Theorem 1.3 is a real application, not decoration.\n\nThe GRH-conditional half of Theorem 1.2 is a clean standard argument: character sums, a contradiction, and a zero-free-region lemma. The unconditional half is sound if you accept estimate (9). I agree with the stress-test note: (9) is the one place where a deep external theorem is doing real work. It is standard and properly cited, but the strength of Theorem 1.2 is exactly the strength of Gallagher's log-free bound, with no extra log factor. A referee should ask the authors to state the precise version and to show it applies to fundamental discriminants in shells of exactly that form. That is a clarification, not a flaw.\n\nSoft spots, in increasing order:\n- Section 3's notation is mangled in the arXiv rendering (β_j, e_j, x_j), making the shell partition hard to verify without reconstruction.\n- Section 5 reports exact values L(4)=570 and L(5)=2679 with no proof or code. These are optional addenda and should be labelled as such or supported.\n- The gap between the unconditional and GRH-conditional exponents is large; the paper does not oversell it, but a reader should be warned.\n\nThe citation pattern is fine. [7] and [8] are used for motivation and comparison; the main theorems do not depend on the authors' earlier work. There is no circularity.\n\nWho it is for: analytic number theorists working on least residues, characters, and binary quadratic forms. It deserves a serious referee. I would send it out.","headline":"A genuinely new finite-field construction drives the lower bound for the least quadratic residue, and the transfer to binary quadratic forms is real progress; the unconditional part rests on an imported zero-density estimate that a referee should check precisely.","tokens_in":11172,"tokens_out":8296,"would_cite":true,"duration_ms":87252,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A15","11E16","11M26","11N05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Given n with k odd prime factors, the least square-free quadratic residue can be as large as 4^{k-o(k)} yet never exceeds C k² 4^k, and the same bounds determine which binary quadratic forms of small discriminant represent every integer up","keywords":["least quadratic residue","quadratic residues modulo n","completely multiplicative functions","finite field F2[x]","binary quadratic forms","fundamental discriminants","zero density estimates","Legendre symbol"],"falsifier":"Fix Δ=10^6, implement Section 3's shell partition, and count how many fundamental discriminants with 1<|D|≤Δ lie in each shell; if any shell contains more than C exp(C e_j) discriminants with the paper's constants, the unconditional N(Δ) bound does not follow. Equivalently, for the square-free n the construction produces, check every fundamental D with |D|≤Δ: a single D with χ_D(p)=1 for all p|n would be a direct counterexample to that instance of Theorem 1.2.","tokens_in":10366,"feed_emoji":"🔢","tokens_out":10109,"duration_ms":97539,"temperature":0.7,"pith_summary":"Let ℓ(n) be the smallest square-free integer r>1 with r≡x² mod n for some x and gcd(r,n)=1. The paper proves that this least quadratic residue is governed by powers of 4: for every n with k odd prime factors, ℓ(n) ≤ C k² 4^k, and for every k≥2 there are infinitely many n for which ℓ(n) ≥ 4^{k−Ck/log k}. Since the number of distinct prime factors of n is at most about log n/log log n, the upper bound becomes ℓ(n) ≤ exp((log 4+O(1/log log n)) log n/log log n), while the constructions give matching lower bounds in terms of n, with a stronger exponent conditional on the Generalized Riemann Hypothesis. Because a modulus n can be arranged so that many small fundamental discriminants are non-residues modulo its prime factors, these bounds translate directly into statements about binary quadratic forms: with discriminant around exp((log 4+o(1)) log N/log log N), every integer up to N is represented by some indefinite form, and there are integers up to N that resist all forms with even slightly smaller discriminants. The interest is that the answer is exponential in k—rather than the 2^k that the density of quadratic residues would naively suggest—and is now known up to the secondary terms.","feed_headline":"Least quadratic residue squeezed between 4^k and k^2 4^k","feed_subtitle":"New bounds fix the exponential order of the smallest square mod n, and show which discriminants let forms represent all integers up to N.","key_machinery":"The lower bound in Theorem 1.1 is carried by a finite-field encoding: primes are assigned irreducible polynomials over F₂[x], and each polynomial's roots in the algebraic closure define a completely multiplicative function f_j(p)=±1. Lemma 2.2, proved through Newton's identity in characteristic 2, says that if z_1,...,z_n are distinct nonzero elements and all odd power sums z_1^{2j−1}+...+z_n^{2j−1} vanish for j=1..k, then n>2^k; applying this to the roots contributed by the prime factors of r forces 2^k < (log r)/log 2 + O(ω(r)), which is exactly why the least r with all f_j(r)=1 must sit near 4^k. For Theorem 1.2 the machinery is instead analytic: fundamental discriminants are partitioned","core_discovery":"The core claim is that the least quadratic residue modulo n has exponential order 4^k in the number k of odd prime factors, and that this phenomenon is stable enough to transfer to binary quadratic forms. Theorem 1.1 supplies the two sides: a pigeonhole argument shows ℓ(n) ≤ C k² 4^k, while a finite-field construction, translating primes into irreducible polynomials over F₂[x], produces moduli n for which any r with ℓ(n)=r must carry more than 2^k distinct algebraic roots, forcing log r ≥ k log 4 − O(k/log k). Theorem 1.2 sharpens this into a discriminant statement: for any Δ one can build an odd square-free n with log n ≤ C log Δ (log log Δ)² (and under GRH log n ≤ C log Δ log log Δ) such t","pith_inferences":["If the finite-field lower bound is genuinely optimal, the eventual answer for general n is ℓ(n) = exp((log 4+o(1)) log n/log log n), meaning the gap to the GRH lower bound reflects the difficulty of the least-nonresidue problem for all characters rather than the true order of ℓ(n).","The construction's resemblance to error-correcting codes suggests a search for explicit moduli n with extremely large ℓ(n) by taking products of primes whose associated F₂ polynomials form a code with large minimum distance; the L(k) values computed in Section 5 are the first few data points.","The same 'kill every small discriminant' strategy could be applied to other families of characters (e.g., cubic or higher-order residues) to produce moduli where the least character value is large, though the binary-quadratic setting exploits that ℓ(n) is itself a fundamental discriminant.","A natural next test is computational: for small Δ, the shell construction can be implemented directly, and the predicted N(Δ) compared with the minimal n found by search, giving a finite check of whether the exponential shell bounds are numerically believable."],"forward_implications":["Every n satisfies ℓ(n) ≤ exp((log 4+O(1/log log n)) log n/log log n), so the growth is exponential in log n/log log n with constant log 4.","There exist n with ℓ(n) ≥ exp(c log n/(log log n)²), so the least quadratic residue can be far larger than any fixed power of log n.","Under GRH, some n have ℓ(n) ≥ exp((log 2/2+o(1)) log n/log log n), narrowing the gap between upper and lower exponential constants.","Every positive integer up to N is representable by a primitive non-degenerate indefinite binary quadratic form of discriminant Δ = exp((log 4+o(1)) log N/log log N); with GRH the same holds for positive-definite forms.","There are integers n≤N that no primitive non-degenerate binary quadratic form with |D| ≤ exp(c log N/(log log N)²) can represent; under GRH this obstruction persists for |D| ≤ exp((log 2/2−ε) log N/log log N)."],"fun_headline_variants":["Least quadratic residue: 4^k is the true order","Smallest square mod n: bounds match up to k^2","Exponential least square residue: 4^k tight","New proof: least QR has order 4^k","Quadratic form application: all ints up to N covered"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The unconditional half of Theorem 1.2 relies on the imported bound that each zero-free shell of fundamental discriminants has size at most C exp(C e_j); if that exponential density estimate were ever exceeded, the constructed small modulus n could have a larger size than claimed.","fun_headline_variants_meta":{"raw":{"variants":["Least quadratic residue: 4^k is the true order","Smallest square mod n: bounds match up to k^2","Exponential least square residue: 4^k tight","New proof: least QR has order 4^k","Quadratic form application: all ints up to N covered"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000962,"raw_usage":{"total_tokens":3899,"prompt_tokens":678,"completion_tokens":3221,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":422,"completion_tokens_details":{"reasoning_tokens":3150}},"tokens_in":422,"tokens_out":3221,"duration_ms":24539,"temperature":1.0,"reasoning_tokens":3150,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T04:26:26.614887+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix Δ=10^6, implement Section 3's shell partition, and count how many fundamental discriminants with 1<|D|≤Δ lie in each shell; if any shell contains more than C exp(C e_j) discriminants with the paper's constants, the unconditional N(Δ) bound does not follow. Equivalently, for the square-free n the construction produces, check every fundamental D with |D|≤Δ: a single D with χ_D(p)=1 for all p|n would be a direct counterexample to that instance of Theorem 1.2.","supporting_citations":[],"review_version":1}