MathAudit

Paper metadata

Original submission
October 19, 2026
Claimed MSC
11B37, 11D09, 11J86
arXiv categories
math.GM
Paper license
CC BY 4.0 ↗
Comments
23 pages

Abstract

In this paper, we discuss two related Diophantine tuple problems involving star numbers Sn=6n(n−1)+1S_n=6n(n-1)+1. First, we investigate classical D(k)D(k)-tuples whose elements are star numbers. We construct infinitely many D(k)D(k)-pairs for several positive and negative admissible values of kk, and we prove several nonexistence results for D(k)D(k)-triples and quadruples. Second, we introduce a modified notion of a star number D(k)D(k)-tuple, requiring aiaj+ka_ia_j+k itself to be a star number. We construct several infinite families of D(1)D(1)-triples, and study their extensions to quadruples. In particular, we prove that {4,108,165}\{4,108,165\} has the unique extension {4,108,165,360}\{4,108,165,360\} to a star number D(1)D(1)-quadruple. The main tools used in this paper are generalized Pell equations, Nagell's theorem, lower bounds for linear forms in logarithms due to Matveev and Dujella-Pethő version of the Baker-Davenport reduction method in diophantine approximation.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated October 23, 2026
01Statements3 reported findingsNo statement issues found

The modular obstructions, infinite Pell families, index separation, and uniqueness of the extension to the star quadruple are supported. Classical square-valued tuples and the modified star-valued tuples are kept distinct in the conclusions.

Theorems 1.4–1.7 and 1.12No issue found

The square-valued star-number pairs and triple obstructions check

PDF pp. 2–3 and 9–13 · arXiv:2610.27033v1

All seven Pell seeds satisfy the stated equations, and multiplication by the positive norm-one unit preserves the parity needed to recover integral star indices. The obstruction for k=11k=11 follows modulo 2727. The index separation follows by placing the product strictly between consecutive squares, with the displayed exceptional pair for k=−4k=-4. For the triple exclusions, the two star residue classes modulo 88 or 99 would have to be pairwise distinct among three entries. In Theorem 1.12 the two Pell solutions force an even difference into a one-integer interval, and substitution gives the final parity contradiction.

Theorems 1.15–1.19No issue found

The modified star-valued triples are integral infinite families

PDF pp. 4–5 and 13–17 · arXiv:2610.27033v1

The first four families reduce their only nonautomatic pair condition to X2−6Y2=−5,−2,3,10X^2-6Y^2=-5,-2,3,10. The recurrence preserves the necessary parity, and also 3∣a3\mid a in the third family and 2∣a2\mid a in the fourth. The other two products have the explicit star indices printed in the proof. Taking differences of the divisibility conditions 12∣ab12\mid ab and 12∣(a+j)b12\mid(a+j)b gives the claimed restrictions on a fourth element. For the pair {4,108}\{4,108\}, the two fundamental Pell classes give exactly the two displayed families.

Theorem 1.20; Corollary 1.21No issue found

The unique fourth element is 360

PDF pp. 17–22 · arXiv:2610.27033v1

The compatibility equations are Y22−27Y12=−26Y_2^2-27Y_1^2=-26 and 55Y22−36Y32=1955Y_2^2-36Y_3^2=19. One Pell class is excluded modulo 88. For the other class, the nonzero logarithmic forms, degree-four height estimates, and continued-fraction reductions bound the remaining indices. Independent high-precision evaluation reproduces the two positive reduction margins 0.1886153629…0.1886153629\ldots and 0.3987344829…0.3987344829\ldots, and bounds m<9.684m<9.684 and m<8.900m<8.900. Exact recurrence evaluation in these finite ranges gives intersections {1,161}\{1,161\} and {1}\{1\}. Excluding a=0a=0 leaves a=(1612−1)/72=360a=(161^2-1)/72=360. Uniqueness then rules out a fifth distinct entry containing the specified triple.

02Proofs3 reported findingsNo proof issues found

The Pell reductions, modular arguments, linear-form estimates, and finite recurrence checks support the central results. A final solution index is misprinted, but its correction is uniquely determined by the displayed initial conditions and does not alter the argument.

Lemmas 2.1–2.4; proofs of Theorems 1.4–1.19No issue found in the reviewed proof

The Pell transformations preserve all reconstruction conditions

PDF pp. 6–7 and 9–17 · arXiv:2610.27033v1

Substitution of Sn=(3(2n−1)2−1)/2S_n=(3(2n-1)^2-1)/2 gives the stated quadratic equations. The unit identities hold exactly, and the recurrences preserve even first coordinates and odd second coordinates where required. The special congruences needed for the rational-looking third elements of the families follow directly from those same recurrences. Positivity and growth give infinitely many distinct admissible elements after discarding the finitely many initial coincidences.

Proof of Theorem 1.20No issue found in the reviewed proof

The logarithmic and finite calculations close the uniqueness proof

PDF pp. 17–22 · arXiv:2610.27033v1

The generalized Pell bounds leave only the displayed fundamental solutions. The two quadratic fields have rational intersection, excluding a zero logarithmic form. The heights permit A1=36A_1=36, A2=8A_2=8, A3=11A_3=11 and the Matveev estimate gives m<1.6⋅1017m<1.6\cdot10^{17}. Both printed rational convergents and both reduction margins were recomputed with 100-digit arithmetic. Exact integer recurrences then check the remaining ranges, including the small initial indices, and recover the asserted unique extension.

Final finite check in Case 3Suspected typo

The index of the common value 161 is 1, rather than 2

PDF p. 20 · final sentence of Case 3 in the proof of Theorem 1.20 · arXiv:2610.27033v1

The initial conditions give ξ1,0′=1\xi'_{1,0}=1 and η2,0′=1\eta'_{2,0}=1, followed by ξ1,1′=η2,1′=161\xi'_{1,1}=\eta'_{2,1}=161. The displayed identification of the common value with index 22 should therefore use index 11 in both sequences. The next terms are 83718371 and 2865728657, respectively. The numerical value, the common-value set, and the resulting fourth element 360360 are unchanged.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper
arXiv:2610.27033v1
Authors listed
Tapas Chatterjee, Sagar Mandal
Audit date
October 23, 2026
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