# P2742: Square-class collisions in the Pell-Lucas sequence

- ID: `P2742`
- Reference: `pell-lucas-squareclass-collisions`
- Page: https://theoremdb.org/statements/P2742
- Record maturity: Reviewed problem with recorded work

## 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.

### Problem setup

- **Remark.** Two positive integers lie in the same square class when their product is a perfect square.
- **Definition.** A perfect square means k^2 for some integer k.

### What 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.

## Status

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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

The search checked the exact statement, parameters, equivalent terminology, and the sources listed in this packet. Database silence is treated only as bounded status evidence. A complete resolution must satisfy every acceptance condition in the canonical problem.

### Background and intake notes

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.

- Original intake status: UNKNOWN as of 2026-07-27. 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.

- Recorded example: The sequence begins 0,1,4,15,56,209,780,2911. Thus (m,n)=(1,2) is a solution because U_1U_2=4.

### Open directions

- **Route 1** (reported): 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. [1](#reference-1)

### 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.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `pell-lucas-squareclass-collisions`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Y. Bilu, G. Hanrot, and P. M. Voutier, “Existence of primitive divisors of Lucas and Lehmer numbers”. Journal für die reine und angewandte Mathematik (Crelles Journal) 2001(539) (2001). DOI 10.1515/crll.2001.080. The contributor selected the Lucas sequence attached to the unit 2+sqrt(3) and formulated the pair-classification target after reviewing primitive-divisor literature. https://doi.org/10.1515/crll.2001.080
   - Also cited at Main primitive-divisor theorem and its exceptional small indices
   - Also cited at Editorial research route recorded 2026-08-01
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - 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.
2. <a id="reference-2"></a>A. Bremner and N. Tzanakis, “On squares in Lucas sequences”. arXiv:math/0610732 (2006). Perfect-square terms in Lucas sequences and the small-index boundary https://arxiv.org/abs/math/0610732
   - preprint; reference source; arXiv:math/0610732v1; checked 2026-08-01
   - Source use: citation_only
   - 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.