# P2744: Minimal vanishing sums of distinct 105th roots

- ID: `P2744`
- Reference: `minimal-vanishing-105th-root-sums`
- Page: https://theoremdb.org/statements/P2744
- Record maturity: Reviewed problem with recorded work

## Problem

Let \(\zeta_{105}=e^{2\pi i/105}\). Classify every subset \(S\subseteq\mathbb Z/105\mathbb Z\) such that \(\sum_{s\in S}\zeta_{105}^s=0\) and no nonempty proper subset of \(S\) has zero sum, up to transformations \(S\mapsto a+uS\) with \(a\in\mathbb Z/105\mathbb Z\) and \(u\in(\mathbb Z/105\mathbb Z)^\times\).

### Problem setup

- **Definition.** Inclusion-minimal means that the displayed sum vanishes and the sum over every nonempty proper subset is nonzero.
- **Remark.** Translation by a rotates all roots by a common 105th root, while multiplication by a unit u applies a Galois automorphism.

### What counts as a solution

- Provide one representative for every affine-Galois orbit of inclusion-minimal subsets S and verify Phi_105 divides each representative polynomial.
- Prove completeness, either structurally or with a machine-checkable exhaustive certificate, and certify inclusion-minimality for every representative.

## Status

Exact enumeration finds seven affine-Galois orbits through weight 19; weights 20 and above, and therefore the complete fixed-conductor classification, remain open.

## Work

### Evidence for the current status

**Computation 1 (There are seven affine-Galois orbits through weight nineteen).** Exact enumeration finds seven affine-Galois orbits through weight 19; weights 20 and above, and therefore the complete fixed-conductor classification, remain open.

Let \(\zeta=\zeta_{105}\), and identify a subset with its exponents in \(\mathbb Z/105\mathbb Z\). The equivalence relation is \(S\sim a+uS\), where \(a\in\mathbb Z/105\mathbb Z\) and \(u\) is a unit modulo 105. Exact enumeration of all nonempty inclusion-minimal vanishing subsets of weight at most 19 gives the following canonical representatives:

- weight 3: \(\{0,35,70\}\);
- weight 5: \(\{0,21,42,63,84\}\);
- weight 7: \(\{0,15,30,45,60,75,90\}\);
- weight 14: \(\{0,1,8,22,29,30,43,45,60,64,71,75,90,92\}\);
- weight 16: \(\{0,1,5,20,22,30,35,43,45,60,64,65,75,80,90,95\}\);
- weight 18: \(\{0,1,3,11,16,24,26,41,42,45,46,61,63,71,76,84,86,87\}\) and \(\{0,1,3,18,22,29,30,33,45,48,60,63,64,71,75,90,92,93\}\).

The corresponding orbit sizes are 35, 21, 15, 105, 105, 315, and 210. Their stabilizer sizes in the affine group of order 5,040 are 144, 240, 336, 48, 48, 16, and 24. No orbit occurs at weights 1, 2, 4, 6, 8 through 13, 15, 17, or 19. The search uses exact remainders modulo \(\Phi_{105}\), exact Boolean constraints, and complete affine-orbit blocking. It is a bounded classification and does not settle whether further orbits occur at weight 20 or above.

### Background and intake notes

A complete orbit catalog would extend weight-bounded tables with a fixed-conductor view. Polynomial remainders, stabilizers, and rejected subcycles remain useful in later conductor classifications.

- Original intake status: UNKNOWN as of 2026-07-27. General structural theorems and classifications through bounded weight are available, but a complete orbit list for distinct 105th roots was not located.
- A 2026-07-27 search checked Lam and Leung, DOI:10.1006/jabr.1999.8089, and Christie, Dykema, and Klep, arXiv:2008.11268. The latter classifies minimal vanishing sums through weight 16 and gives computation through weight 21, not a complete fixed-modulus list.
- The modulus 105=3*5*7 is the smallest odd squarefree modulus with three prime factors. It is the first such modulus where prime cycles of three different lengths can interact.
- Represent S by its 0-1 polynomial f_S(X). Vanishing is equivalent to Phi_105(X) dividing f_S(X) over Q, so exact polynomial remainders and canonical orbit representatives provide compact certificates.
- Trap: allowing repeated roots changes the object from subsets to multisets. Minimality must be checked against every nonempty proper subset.

- Recorded example: S={0,35,70} is inclusion-minimal because 1+zeta_105^35+zeta_105^70=0 and no one- or two-term subsum vanishes.

### Other known results

- **Claim 1** (supported): Lam and Leung's construction, lower bound, and uniqueness theorem give one rotation class of asymmetric minimal sums at the smallest possible weight 14 for primes 3, 5, and 7. [2](#reference-2)
- **Claim 2** (supported): Christie, Dykema, and Klep prove the type classification through weight 16; the current revision labels the implemented extension through weight 21 conjectural. [1](#reference-1)

### Prior approaches

- **Route 1** (reproduced): Sequential exact solving, orbit blocking, proper-subset scans, and independent matrix construction agree on seven orbits and certify every omitted weight through 19.
- **Route 2** (supported): A dated audit found strong type and minimum-weight results, corrected a source-version trap, and found no primary source giving the full fixed-conductor affine classification. [1](#reference-1) [2](#reference-2) [3](#reference-3)

### Open directions

- **Route 3** (reported): Continue the same exact matrix search at weight 20 with every orbit through weight 19 installed as a minimal-subset blocker, then independently replay any survivors.

### Runnable artifacts

- **Artifact 1** (reproduced): A deterministic Python, SymPy, and Z3 program reconstructs the 48 by 105 cyclotomic matrix and exhaustively enumerates the seven minimal affine-Galois orbits through weight 19.

### Computational notes

- Exact polynomial division verifies that 1+X^35+X^70 is divisible by Phi_105(X). No complete enumeration is claimed.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `minimal-vanishing-105th-root-sums`, 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>Louis Christie, Kenneth J. Dykema, and Igor Klep, “Classifying minimal vanishing sums of roots of unity”. arXiv:2008.11268 (2020). Louis Christie, Kenneth J. Dykema, and Igor Klep, Classifying minimal vanishing sums of roots of unity, arXiv:2008.11268v2, Definition of relative order, Theorem 3.3, Table 1, Section 4, and Appendix B; arXiv:math/9511209v1; arXiv:2008.11268v1 and v2; GitHub repository lchristie/Sums-of-Roots-of-Unity at commit b0563b270dc89ed7ca9e195535a107d5a81cc6dc; production audit dated 2026-07-28 UTC; Definition of relative order, Theorem 3.3, Table 1, Section 4, Appendix B, and submission history https://arxiv.org/abs/2008.11268
   - Also cited at Definition of relative order, Theorem 3.3, Table 1, Section 4, Appendix B, and submission history
   - Also cited at Louis Christie, Kenneth J. Dykema, and Igor Klep, Classifying minimal vanishing sums of roots of unity, arXiv:2008.11268v2, Definition of relative order, Theorem 3.3, Table 1, Section 4, and Appendix B
   - preprint; reference source; arXiv:2008.11268, version checked 2026-07-28; checked 2026-07-28
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For Minimal vanishing sums of distinct 105th roots: The proved type classification stops at weight sixteen. Christie, Dykema, and Klep prove the type classification through weight 16; the current revision labels the implemented extension through weight 21 conjectural. Audit the literature and current TheoremDB state. A dated audit found strong type and minimum-weight results, corrected a source-version trap, and found no primary source giving the full fixed-conductor affine classification.
2. <a id="reference-2"></a>T.Y Lam and K.H Leung, “On Vanishing Sums of Roots of Unity”. Journal of Algebra 224(1) (2000), 91-109. DOI 10.1006/jabr.1999.8089. T. Y. Lam and K. H. Leung, On vanishing sums for roots of unity, Example 2.5 on pp. 5-6, Lower Bound Theorem 4.8 on pp. 11-12, and Uniqueness Theorem 6.5 on p. 15; J. Algebra 224 (2000), DOI 10.1006/jabr.1999.8089; Example 2.5, Lower Bound Theorem 4.8, Uniqueness Theorem 6.5 https://doi.org/10.1006/jabr.1999.8089
   - Also cited at Example 2.5, Lower Bound Theorem 4.8, Uniqueness Theorem 6.5
   - Also cited at T. Y. Lam and K. H. Leung, On vanishing sums for roots of unity, Example 2.5 on pp. 5-6, Lower Bound Theorem 4.8 on pp. 11-12, and Uniqueness Theorem 6.5 on p. 15; J. Algebra 224 (2000), DOI 10.1006/jabr.1999.8089
   - scholarly_publication; reference source; arXiv:math/9511209, version checked 2026-07-28; checked 2026-08-01
   - Open copy: https://arxiv.org/abs/math/9511209
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For Minimal vanishing sums of distinct 105th roots: The dated search found general and weight-bounded classifications but no complete distinct-root classification for conductor 105.
3. <a id="reference-3"></a>Sums-of-Roots-of-Unity. github.com checked 2026-08-01. repository HEAD b0563b270dc89ed7ca9e195535a107d5a81cc6dc https://github.com/lchristie/Sums-of-Roots-of-Unity
   - software; software source; commit b0563b270dc89ed7ca9e195535a107d5a81cc6dc; checked 2026-07-28
   - Source use: original_summary
   - For Minimal vanishing sums of distinct 105th roots: Audit the literature and current TheoremDB state. A dated audit found strong type and minimum-weight results, corrected a source-version trap, and found no primary source giving the full fixed-conductor affine classification.