# P2430: Merging of orbits under adding the product of nonzero digits

- ID: `P2430`
- Reference: `digit-product-iteration-trajectories`
- Page: https://theoremdb.org/statements/P2430
- Record maturity: Reviewed problem with recorded work

## Problem

Define \(T(n) = n + p(n)\), where \(p(n)\) is the product of the nonzero decimal digits of \(n\). Do all orbits of \(T\) eventually merge into finitely many trajectories, in the sense that there is a finite set \(S\) of orbits such that every starting value eventually enters one of them?

### Remarks

- **Remark.** The product of nonzero digits ignores any digit equal to 0, so p(105) = 1 times 5 = 5.
- **Remark.** Two orbits merge when they share a common value from some point on.

### What counts as a solution

- Prove that only finitely many distinct eventual trajectories occur, or show that infinitely many pairwise disjoint orbits exist.

## Status

The published conjecture gives a single eventual trajectory, which is stronger than the candidate's finite-trajectory question. [3](#reference-3)

## Work

### Evidence for the current status

**Conjecture 1 (Loomis conjectured that every orbit joins the orbit of 1).** The published conjecture gives a single eventual trajectory, which is stronger than the candidate's finite-trajectory question.

For \(T(n)=n+p(n)\), with zero digits omitted from the product, Paul Loomis conjectured that every positive starting value eventually enters the orbit beginning at 1. This exact map and conjecture appeared in his article on digit product sequences and on the companion page's section "Sequences That Join." The candidate independently rediscovered a weaker consequence. The reviewed sources still describe universal joining as a conjecture.

### Background and intake notes

Multiples of 10 with a single nonzero digit behave differently from generic values, since p(n) is then small relative to n, which is the obvious obstruction to a clean merging argument.

- Original intake status: Status not established. No literature search was performed. Maps of the form n plus a digit function are common in recreational number theory and this one may be known.
- No literature search was carried out.
- The map is strictly increasing, so there are no cycles and the question is purely about coalescence of orbits, not about eventual periodicity.

- Recorded example: The orbit of 1 begins 1, 2, 4, 8, 16, 22, 26, 38, 62, 74, 102, 104, 108, 116, 122.

### Other known results

- **Proposition 1** (supported): OEIS A063114 records the map, while A063108 records the orbit of 1. [2](#reference-2)

### Open directions

- **Conjecture 1** (supported): The published conjecture gives a single eventual trajectory, which is stronger than the candidate's finite-trajectory question. [3](#reference-3)
- **Route 1** (conjectured): Universal joining to the orbit of 1 implies the candidate, while the weaker finite-trajectory statement may admit a separate proof. [2](#reference-2)

### Runnable artifacts

- **Artifact 1** (supported): Every start through one million was reported to join the orbit of 1. [1](#reference-1)
- **Artifact 2** (reproduced): The 199 starts produce 31 distinct values after exactly 60 applications of the map.

### Computational notes

- Iterating 60 steps from each of the 199 starting values 1 through 199 produced only 31 distinct values, so the great majority of these orbits had already coalesced.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `digit-product-iteration-trajectories`, 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>Paul Loomis, Integer Sequences, section III, Sequences That Join; cross-referenced by OEIS A096287. facstaff.bloomu.edu checked 2026-08-01. Paul Loomis, Integer Sequences, section III, Sequences That Join; cross-referenced by OEIS A096287 https://facstaff.bloomu.edu/ploomis/sequences.html
   - website; primary source; web version checked 2026-08-01; checked 2026-08-01
   - Source use: original_summary
   - Reports that every starting value through one million joins the canonical digit-product trajectory.
2. <a id="reference-2"></a>Open direction relative to the conjecture recorded by Loomis and OEIS A063108. OEIS entry A063108, checked 2026-08-01. Open direction relative to the conjecture recorded by Loomis and OEIS A063108 https://oeis.org/A063108
   - Also cited at OEIS A063108, A063114, and A096287
   - reference_database; primary source; web version checked 2026-08-01; checked 2026-08-01
   - Source use: original_summary
   - Records the canonical digit-product trajectory and its linked map and joining data.
3. <a id="reference-3"></a>Paul A. Loomis, An Introduction to Digit Product Sequences, Journal of Recreational Mathematics 32 (2003-2004), pages 147-151; companion page section III. Paul A. Loomis, An Introduction to Digit Product Sequences, Journal of Recreational Mathematics 32 (2003-2004), pages 147-151; companion page section III https://plouffe.fr/OEIS/citations/itseq4.pdf
   - journal_article; primary source; PDF checked 2026-08-01; checked 2026-08-01
   - Source use: original_summary
   - Introduces the digit-product iteration and conjectures that every start joins the orbit of 1.
   - Source named by the research packet.