{"id":"eda36ab7-81ce-45a6-ac1a-70fefa4f06a6","arxiv_id":"2506.04087","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For infinitely many primes p there is a choice of c such that no square of side sqrt(0.1 log p) contains two points, giving a sqrt(log p) lower bound for the guaranteed side length.","lead":"This paper studies the modular hyperbola xy ≡ c (mod p), the set of integer points on a discrete hyperbola over a prime field. It finds how large a square must be to be guaranteed to contain two of these points, and what happens when the horizontal and vertical distances between them are restricted to special sets such as primes, squarefree numbers, or smooth numbers.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1 hinges on an unverified residue-class version of the Graham–Ringrose lower bound; confirm that [5] supplies infinitely many primes p ≡ 1 (mod 4).","rationale":"I agree with the reader that the single most load-bearing point for the headline result is the external Graham–Ringrose input. I checked the internal character-sum step of Theorem 1 carefully: the expansion of the sum Sigma, the use of Weil's bound on products of distinct linear factors, the dominance of p/2^{2L} over the error term when L = 0.1 log p, and the reduction from condition (5) to the Legendre-symbol obstruction for all h, k all appear sound once n_p > L. The Pólya–Vinogradov upper bound mentioned in the proof is not actually needed for the lower bound. The Z0 issue in Theorem 2 is a separate, fixable gap (Z0 should be chosen inside the multiplicatively closed set M), but it does not affect the central claim of Theorem 1. Since the reader's verdict is already CONDITIONAL and the remaining question is verifying the residue-class form of [5], I recommend no change to the verdict: it should remain conditional on confirming that Graham–Ringrose applies to infinitely many primes p ≡ 1 (mod 4) and on repairing the Z0 subset point in Theorem 2.","tokens_in":6192,"tokens_out":23377,"duration_ms":232758,"concrete_test":"Open Graham–Ringrose [5] and locate the theorem giving n_p >> log p log log log p. Determine whether it is stated for infinitely many primes p ≡ 1 (mod 4), or whether the proof can be rerun with the additional condition p ≡ 1 (mod 4) (for example, by applying the same large-sieve/character argument to primes in the progression 1 mod 4). If the theorem does not cover this residue class, Theorem 1 lacks its required infinite family of primes.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1 (Section 3) is internally consistent: the character-sum estimate yields, for each p with n_p > L = floor(0.1 log p) and p sufficiently large, a c in (L/4, p - L/4) for which every l ± 4c with 1 ≤ l ≤ L is a quadratic nonresidue. Since L < n_p implies every positive integer up to L is a quadratic residue, each product hk with |h|, |k| ≤ sqrt(L) is a residue, and therefore (hk/p)(hk - 4c/p) = -1 for all such h, k. The entire infinite-family conclusion therefore rests on the assertion that Graham and Ringrose provide infinitely many such primes in the specific congruence class p ≡ 1 (mod 4). The manuscript cites [5] but does not quote the exact theorem. If the lower bound in [5] is stated for primes in an unspecified residue class, or for a class that cannot be chosen to be 1 (mod 4), then the infinite family in Theorem 1 is unsupported even though the character-sum reduction itself checks out.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the distribution of points on the modular hyperbola xy ≡ c (mod p), specifically the side length H needed to force a square B_{X,Y}(H) to contain at least two points. Theorem 1 gives infinitely many primes p ≡ 1 (mod 4) for which some c satisfies that every square of side about sqrt(0.1 log p) contains at most one point, providing a lower bound of order sqrt(log p). Theorems 2 and 3 give upper bounds of the form p^{1/4} exp((log p)^{1/2+ε}) when the two distances h and k are required to lie in a multiplicatively closed set or in the squarefree numbers, and Theorem 4 gives an exponent 11/34 + ε for distances in an almost dense set. The proofs use character sums, Weil's bound, Burgess's moment method, and a double character sum bound of Karatsuba and Chang.","tokens_in":6330,"tokens_out":40408,"duration_ms":344118,"significance":"If the proofs are correct, Theorem 1 is a genuine improvement in the study of small squares on modular hyperbolas, providing the first lower bound of order sqrt(log p) for the side length needed to force two points for all c. The paper also gives a nice framework for distance-constrained variants, and the character-sum reductions in Section 2 are elegant and self-contained. The author is explicit about the standard character-sum ingredients, and the subexponential factors in Theorems 2 and 3 are designed to cancel correctly. However, the presentation of the proofs of Theorems 2 and 3 is sufficiently compressed that some steps are ambiguous, and one step in the reduction appears to involve the inverse of h rather than h itself.","major_comments":[{"comment":"Theorem 1 depends on the assertion that Graham and Ringrose [5] supply infinitely many primes p ≡ 1 (mod 4) with n_p ≫ log p log log log p. The manuscript cites [5] without quoting the theorem, so the reader cannot verify that the required congruence class is covered. This is load-bearing because without infinitely many such primes in the class p ≡ 1 (mod 4), the infinite family in Theorem 1 would not follow. Please state the exact Graham–Ringrose theorem used and confirm that it applies to p ≡ 1 (mod 4); the standard form of that theorem does provide such primes, so this should be an explicit correction rather than a change of result.","section":"Section 3, final paragraph"},{"comment":"The character sum S is written as Σ_{h∈A,z0∈Z0,z1∈Z1} χ(z0 − 4c h z1). The vanishing condition for this summand is h z0 z1 ≡ 4c (mod p), which corresponds to a pair with one distance equal to h z1. Since both h and z1 range up to H, the product h z1 can be as large as H^2, and H^2 = p^{1/2} exp((log p)^{1/2+ε}) is much larger than the theorem's claimed bound p^{1/4} exp((log p)^{1/2+ε}). As written, the argument therefore appears to prove the existence of two points whose one distance is bounded by H^2, not by H, and this would be a strictly weaker statement than Theorem 2. The same issue affects the proof of Theorem 3 in Section 5. Please clarify which variables are the final h and k, or use a character sum of the form χ(z0 z1 − 4c \\bar h) whose vanishing condition is h z0 z1 ≡ 4c mod p, matching the claimed bound h ≤ H, k ≤ H T.","section":"Section 4, around Eqs. (6)–(7)"}],"minor_comments":[{"comment":"The second character condition should be (k/p)(k − 4c \\bar h / p) = 1, with the multiplicative inverse of h, rather than k − 4ch; the first displayed form hk(hk − 4c) is correct, but the second form as written is missing the inverse. The same typo appears in Eq. (6) in Section 4.","section":"Section 2, Eq. (3)"},{"comment":"The phrase 'multiplicative closed set' should be 'multiplicatively closed set' in Definition 1 and in the statements of Theorems 2 and 3.","section":"Introduction, Definitions"},{"comment":"The sentence 'This together with (1) and (3)' appears to refer to the reduction (3), not the earlier theorem labelled (1); please correct the cross-reference.","section":"Section 3, proof of Theorem 1"},{"comment":"The notation 'gcd(c_p, p) = 1' is awkward; c is an integer, and the subscript in c_p appears only in the statement. Please use a single symbol consistently.","section":"Theorem 1 statement"},{"comment":"The word 'subet' should be 'subset'. Also, the 'almost dense' condition is stated only on dyadic intervals [X,2X] but is applied to the initial interval [1,p^α]; the deduction uses a standard dyadic summation and should be made explicit.","section":"Section 6, Theorem 4 proof"}],"recommendation":"major_revision","confidential_remarks":"The core ideas appear sound and the character-sum reductions are promising, but the proof of Theorems 2 and 3 has a notational/structural ambiguity that is load-bearing: the variables in the estimated character sum do not obviously match the distances claimed in the theorem. The author should be asked to rewrite that part with explicit definitions of the final h and k. The Graham–Ringrose citation issue is easily fixed by quoting the theorem. I would not recommend rejection on the basis of these issues, since they seem repairable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Chan's paper has more in it than the title suggests: Theorem 1 is the first lower bound of order sqrt(log p) on the side length of a square that can contain two points on the modular hyperbola, for infinitely many primes, and Theorems 2-4 give genuinely new upper bounds when the two coordinates differ by numbers of special shape. The character-sum machinery is standard—Weil, Burgess, Karatsuba/Chang—but the reductions are clean and the applications are honest.\n\nThe main derivations check out. The reduction of two points to the product (hk/p)(hk-4c/p) is simple and correct, and the use of the Burgess moment method in Theorems 2 and 3 is fine; the subexponential factors cancel as advertised. Theorem 4's application of Chang's double character sum condition with k=3 works numerically. The self-citation [1] is used for context and the transformation is re-proven in Section 2, so no circularity.\n\nThe soft spots are real but fixable. In Theorem 2, the set Z0 is chosen as the larger of the quadratic residues or nonresidues in [1,T], without any intersection with M, despite the text claiming 'by pigeonhole principle and positive lower density of M.' As written, the proof does not guarantee that z0 z1 lies in M, so the contradiction to (6) is not licensed. The fix is to take Z0 as the larger of M∩[1,T]∩Qp or M∩[1,T]\\Qp, which has size ≫δ T by the density assumption. This looks like an oversight rather than a deep flaw.\n\nThe bigger external dependency is Theorem 1. The character-sum estimate gives a c with the required nonresidue property once n_p > 0.1 log p, and Pólya–Vinogradov supplies n_p ≪ sqrt(p) log p, so the width of the interval works. But the infinite family relies entirely on Graham–Ringrose yielding infinitely many primes p ≡ 1 mod 4 with n_p ≫ log p log log log p. The paper cites [5] without quoting the statement. I believe their theorem is usually stated for all primes, and the construction can be adapted to p ≡ 1 mod 4, but this needs to be verified and the exact statement included. The stress-test note overstates it as 'unverified'; it's a citation hygiene issue unless the GR theorem truly doesn't cover the class.\n\nOverall: the paper is a solid extension in a niche area, not a breakthrough. It deserves a serious referee, and I'd expect acceptance after a minor revision that fixes the Z0 definition and quotes the Graham–Ringrose theorem precisely.","headline":"Solid, well-executed extension in a niche area: new lower and upper bounds for close points on a modular hyperbola, with two fixable issues (Theorem 2's Z0 definition and the precise Graham–Ringrose statement).","tokens_in":6917,"tokens_out":6426,"would_cite":true,"duration_ms":53889,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11L40","11T24","11A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a first lower bound for the side of a square that must contain two points of a modular hyperbola: for infinitely many primes $p \\equiv 1 \\pmod{4}$ there is a residue $c$ modulo $p$ such that every square of side $\\lfloor…","keywords":["modular hyperbola","character sums","Legendre symbol","quadratic nonresidues","squarefree numbers","smooth numbers","almost dense sets","lower bounds"],"falsifier":"Compute, for primes $p \\equiv 1 \\pmod{4}$ up to a large bound, the quantity $M(p) = \\max_c m(p,c)$, where $m(p,c)$ is the least $H$ for which some square of side $H$ contains two points of $xy \\equiv c \\pmod{p}$. If $M(p) \\le \\lfloor \\sqrt{0.1\\log p}\\rfloor$ for every tested prime, the infinite family promised by Theorem 1 would not appear in the data; a single prime with $M(p) > \\lfloor \\sqrt{0.1\\log p}\\rfloor$ would exhibit the claimed phenomenon and validate the counting mechanism.","tokens_in":5918,"feed_emoji":"🔢","tokens_out":19429,"duration_ms":176984,"temperature":0.7,"pith_summary":"The paper studies solutions of $xy \\equiv c \\pmod{p}$, the points of a modular hyperbola, and asks how large a square must be before it is guaranteed to contain two of them. Its main result is a lower bound: for infinitely many primes $p \\equiv 1 \\pmod{4}$ there is a choice of $c$ for which every square of side $\\lfloor \\sqrt{0.1\\log p}\\rfloor$ still contains at most one point. This is the first lower bound of order $\\sqrt{\\log p}$ on the side length needed to force two points, and it comes from counting, with Weil's bound on character sums, values of $c$ that avoid small quadratic residues. The paper also proves upper bounds on the side length when the gaps between the two points must belong to special sets: multiplicatively closed sets, squarefree numbers, and almost dense sets such as the primes.","feed_headline":"Hyperbola pairs stay apart in squares as wide as sqrt(log p)","feed_subtitle":"A first lower bound: for infinitely many primes, some hyperbola keeps every square of side sqrt(log p) at one point.","key_machinery":"The load-bearing identity is the Legendre-symbol criterion (3): two points $(x,y)$ and $(x+h,y+k)$ lie on $xy \\equiv c \\pmod{p}$ exactly when $\\left(\\frac{k}{p}\\right)\\!\\left(\\frac{k-4c\\bar{h}}{p}\\right)=1$ for some $1 \\le h,k \\le H$, where $\\left(\\frac{a}{p}\\right)$ is $1$, $-1$, or $0$ according as $a$ is a square, a non-square, or $0$ modulo $p$. This converts the geometry of small squares into a character-sum counting problem. For Theorem 1, the counting is done with Weil's bound on products of Legendre symbols, combined with the Graham-Ringrose lower bound on the least quadratic nonresidue $n_p$, which guarantees that the interval of admissible $c$ is nonempty for infinitely many $p \\equiv 1 \\pmod{4}$. For Theorems 2 and 3, the same criterion is fed through Burgess-type estimates and Hölder's inequality with dense sets of $h$ and $k$; for Theorem 4, a double character-sum bound of Karatsuba type, in the sharpened form due to Chang, supplies the estimate.","core_discovery":"The central claim is Theorem 1: there are infinitely many primes $p \\equiv 1 \\pmod{4}$ and integers $c$ with $\\gcd(c,p)=1$ such that every square of side $\\lfloor \\sqrt{0.1\\log p}\\rfloor$ contains at most one point of the modular hyperbola $xy \\equiv c \\pmod{p}$. In other words, no universal guarantee of two points can be given with side length $o(\\sqrt{\\log p})$. The proof reduces the existence of two points $(x,y)$ and $(x+h,y+k)$ on the hyperbola to the condition that the Legendre symbols satisfy $\\left(\\frac{k}{p}\\right)\\!\\left(\\frac{k-4c\\bar{h}}{p}\\right)=1$ for some $1 \\le h,k \\le H$, then uses Weil's bound to count residues $c$ for which all small $h,k$ fail this test. The remaining theorems give upper bounds for restricted gaps: $H = C_{\\delta,\\epsilon}\\,p^{1/4}\\exp((\\log p)^{1/2+\\epsilon})$ for a multiplicatively closed set of positive density, the same shape for squarefree gaps (with one gap prime and the other a product of two primes as an option), and $H = C_{\\epsilon,A}\\,p^{11/34+\\epsilon}$ for any almost dense set $A$, which covers prime gaps.","pith_inferences":["Extending the argument: the true threshold for the side length that forces two points for every $c$ lies somewhere between $\\sqrt{\\log p}$ and $p^{1/4}$, and the paper's lower bound suggests the answer may be a small power of $\\log p$ rather than a power of $p$.","Because Theorem 1 inherits the Graham-Ringrose input, any improvement in unconditional lower bounds for least quadratic nonresidues in the residue class $1 \\pmod{4}$ would immediately enlarge the constant $0.1$ and possibly extend the result to all primes.","The subexponential saving in Theorems 2 and 3 hints that the obstruction in the earlier $p^{1/4+\\epsilon}$ result is concentrated on a thin set of bad gap pairs, so other naturally defined sets of gaps should admit similar savings through the same character-sum route."],"forward_implications":["No universal guarantee can force two points on $xy \\equiv c \\pmod{p}$ for every $c$ with a square of side $o(\\sqrt{\\log p})$; Theorem 1 puts a hard floor on the side length for an infinite family of primes.","For any multiplicatively closed set of positive density, two hyperbola points whose horizontal and vertical gaps lie in the set can be found in a square of side $p^{1/4}\\exp((\\log p)^{1/2+\\epsilon})$, improving the earlier $p^{\\epsilon}$ loss to a subexponential factor.","The same upper bound holds when the gaps are squarefree, and in that case one gap can even be taken prime while the other is a product of two distinct primes.","For any almost dense set, including the primes, two points with gaps in the set are guaranteed inside a square of side $p^{11/34+\\epsilon}$."],"supporting_citations":[{"why":"Establishes the earlier $p^{1/4+\\epsilon}$ guarantee and the reduction to the character-sum condition (3) that this paper extends.","marker":"[1]"},{"why":"Supplies Weil's bound on multiplicative character sums (Theorem 5), used to estimate sums in Theorems 1–3.","marker":"[3]"},{"why":"Provides the lower bound on least quadratic nonresidues for infinitely many primes $p \\equiv 1 \\pmod{4}$ on which Theorem 1 depends.","marker":"[5]"},{"why":"Gives the sharpened double character sum bound (condition (13)) used to prove Theorem 4.","marker":"[2]"},{"why":"Provides the original double character sum estimate (condition (12)) whose sharpening by Chang is applied in Theorem 4.","marker":"[4]"}],"fun_headline_variants":["Infinitely many primes have a hyperbola where sqrt(log p) squares are one-point","For many primes, modular hyperbola keeps sqrt(log p) squares point-sparse","sqrt(log p) side squares can avoid two hyperbola points for infinitely many primes","New bound: hyperbola pairs cannot be forced in sqrt(log p) squares"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument leans on a theorem of Graham and Ringrose, quoted without proof here, that infinitely many primes $p \\equiv 1 \\pmod{4}$ have a smallest non-square modulo $p$ at least of size $\\log p\\,\\log\\log\\log p$; if that result were not available for this residue class, Theorem 1 would not follow from the given proof.","fun_headline_variants_meta":{"raw":{"variants":["Infinitely many primes have a hyperbola where sqrt(log p) squares are one-point","For many primes, modular hyperbola keeps sqrt(log p) squares point-sparse","sqrt(log p) side squares can avoid two hyperbola points for infinitely many primes","New bound: hyperbola pairs cannot be forced in sqrt(log p) squares"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001379,"raw_usage":{"total_tokens":5562,"prompt_tokens":895,"completion_tokens":4667,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":511,"completion_tokens_details":{"reasoning_tokens":4580}},"tokens_in":511,"tokens_out":4667,"duration_ms":42045,"temperature":1.0,"reasoning_tokens":4580,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:50:19.148866+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for primes $p \\equiv 1 \\pmod{4}$ up to a large bound, the quantity $M(p) = \\max_c m(p,c)$, where $m(p,c)$ is the least $H$ for which some square of side $H$ contains two points of $xy \\equiv c \\pmod{p}$. If $M(p) \\le \\lfloor \\sqrt{0.1\\log p}\\rfloor$ for every tested prime, the infinite family promised by Theorem 1 would not appear in the data; a single prime with $M(p) > \\lfloor \\sqrt{0.1\\log p}\\rfloor$ would exhibit the claimed phenomenon and validate the counting mechanism.","supporting_citations":[{"cited_title":"Banks, M.Z","cited_arxiv_id":null,"evidence_quote":"Establishes the earlier $p^{1/4+\\epsilon}$ guarantee and the reduction to the character-sum condition (3) that this paper extends."},{"cited_title":"Chang, On a question of Davenport and Lewis and new character sum bounds in finite fields, Duke Math","cited_arxiv_id":null,"evidence_quote":"Supplies Weil's bound on multiplicative character sums (Theorem 5), used to estimate sums in Theorems 1–3."},{"cited_title":"Iwaniec and E","cited_arxiv_id":null,"evidence_quote":"Provides the lower bound on least quadratic nonresidues for infinitely many primes $p \\equiv 1 \\pmod{4}$ on which Theorem 1 depends."},{"cited_title":"Chan, Shortest distance in modular hyperbola and least quadratic non-residue, Mathematika 62 (2016), 860--865","cited_arxiv_id":null,"evidence_quote":"Gives the sharpened double character sum bound (condition (13)) used to prove Theorem 4."},{"cited_title":"Granville and K","cited_arxiv_id":null,"evidence_quote":"Provides the original double character sum estimate (condition (12)) whose sharpening by Chang is applied in Theorem 4."}],"review_version":1}