[#P2742] Square-class collisions in the Pell-Lucas sequence
Problem. Let \(U_0=0\), \(U_1=1\), and \(U_{n+2}=4U_{n+1}-U_n\) for \(n\ge0\). Determine all pairs of integers \(1\le m<n\) for which \(U_mU_n\) is a perfect square.
1Context
This is a square-class problem rather than a termwise perfect-power problem. Its intermediate valuation lemmas and primitive-divisor exclusions should transfer to other Lucas sequences.
2Definitions
Definition 1 (Two positive integers lie in the same square class when their product). Two positive integers lie in the same square class when their product is a perfect square.
Definition 2 (A perfect square). A perfect square means k^2 for some integer k.
3What counts as a solution
- List every pair (m,n) with 1<=m<n and U_mU_n square, and prove that no other pair exists.
- Any appeal to a primitive-divisor theorem must separately handle every excluded small index and must track valuation parity, not only the presence of a new prime.
1Status
Current status (Current status and unresolved remainder). UNKNOWN as of 2026-08-01. Primitive-divisor theorems sharply constrain large indices in Lucas sequences, but the exact pair classification for this recurrence was not located in the checked sources.[1]
1Records
Notes and companion material
Original intake status. OPEN as of 2026-08-01. UNKNOWN as of 2026-08-01. Primitive-divisor theorems sharply constrain large indices in Lucas sequences, but the exact pair classification for this recurrence was not located in the checked sources.
- A 2026-07-27 search checked Bilu, Hanrot, and Voutier on primitive divisors of Lucas and Lehmer numbers and searches for the exact recurrence together with square products. No exact classification was located.
- The coefficient 4 is natural because the characteristic roots are 2+sqrt(3) and 2-sqrt(3), the fundamental positive units associated with x^2-3y^2=1.
- Strong divisibility gives gcd(U_m,U_n)=U_gcd(m,n). A proof should retain primitive-prime valuations and index divisibility, since these lemmas can be reused for other recurrence pairs.
- Trap: checking whether U_m and U_n are separately squares loses solutions where their nonsquare parts agree.
- Fresh exact-title, parameter, source, and corpus searches were completed on 2026-08-01.
Recorded example 1. The sequence begins 0,1,4,15,56,209,780,2911. Thus (m,n)=(1,2) is a solution because U_1U_2=4.
Computational notes
- Exact factorization through n=40 found (1,2) as the only pair in that range whose product is a square. The bounded factorization is evidence only.
How the 2 records connect
ProblemSquare-class collisions in the Pell-Lucas sequence
2See also
How to cite
TheoremDB contributors, “Square-class collisions in the Pell-Lucas sequence,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/pell-lucas-squareclass-collisionsThis page as plain text: pell-lucas-squareclass-collisions.md
This problem includes 2 records joined by 1 typed links, sourced from doi.org[1], current as of August 1, 2026.
1References
- Packet source. Yann Bilu, Guillaume Hanrot, and Paul M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, Journal für die reine und angewandte Mathematik 539 (2001), 75-122. Main primitive-divisor theorem and its exceptional small indices. ↗journal article · primary source · checked 2026-08-01Source use: original summary.This is the primary or maintained source used to check the formulation, neighboring results, and current research boundary.Also cited at The contributor selected the Lucas sequence attached to the unit 2+sqrt(3) and formulated the pair-classification target after reviewing primitive-divisor literature.Also cited at Editorial research route recorded 2026-08-01.Source used to assess the problem's recorded status.For Square-class collisions in the Pell-Lucas sequence, the reviewed source scope is The contributor selected the Lucas sequence attached to the unit 2+sqrt(3) and formulated the pair-classification target after reviewing primitive-divisor literature.. The packet makes no inference beyond that cited scope.Source named by the research packet.
- A. Bremner and N. Tzanakis, On squares in Lucas sequences, 2006, arXiv:math/0610732v1. Perfect-square terms in Lucas sequences and the small-index boundary. ↗preprint · secondary source · arXiv:math/0610732v1 · checked 2026-08-01Source use: original summary.This later or complementary source was checked for equivalent formulations, methods, and possible prior answers.Source used to assess the problem's recorded status.For Square-class collisions in the Pell-Lucas sequence: This later or complementary source was checked for equivalent formulations, methods, and possible prior answers.
CC0 square-class collision problem for a natural nondegenerate Lucas sequence.