# P2630: Simultaneous additive and multiplicative Sidon sets in F_101

- ID: `P2630`
- Reference: `simultaneous-sum-product-sidon-f101`
- Page: https://theoremdb.org/statements/P2630
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/simultaneous-sum-product-sidon-f101?representation=markdown
- Record maturity: Prospecting problem with recorded work

## The problem

Determine the largest subset \(A\subseteq\mathbb F_{101}^{*}\) such that the sums \(a+b\) are distinct over all unordered pairs with repetition \(a\le b\) in \(A\), and the products \(ab\) are separately distinct over the same pairs.

### Remarks

- **Remark.** The ordering a<=b uses the standard representatives 1 through 100 only to name each unordered pair.
- **Remark.** A collision among two sums or among two products is forbidden. A sum may equal a product.

### What counts as a solution

- Give a valid set attaining the maximum and a complete branch-and-bound, SAT, or equivalent certificate excluding the next size.

## Status

**Resolution status:** Solution submitted.
**Lean verification:** Not Lean-verified.

A nine-element set meets both Sidon conditions, while the multiplicative condition alone excludes size 10. [1](#reference-1)

**Recorded scope.** subsets of the nonzero elements of F_101 whose unordered pair sums and unordered pair products, with repetition, are separately distinct

[Read this exact record and its supporting evidence](/records/sps101-claim-exact-maximum-nine)


### Verification

The answer is
\[
\max |A|=9.
\]
A witness is
\[
A=\{1,11,47,62,67,70,71,84,96\}.
\]
Its 45 unordered pair sums with repetition are distinct in \(\mathbb F_{101}\), and its 45 corresponding products are also distinct.

For the upper bound, 2 generates \(\mathbb F_{101}^{*}\), which has order 100. Write every element of a product-Sidon set as \(2^b\), with \(b\in\mathbf Z/100\mathbf Z\). Distinct pair products are equivalent to distinct pair sums of the exponents. This is also equivalent to requiring all directed differences \(b_i-b_j\), for \(i\ne j\), to be distinct. Thus a size-10 product-Sidon set would give a \((100,10)\)-modular Golomb ruler.

Buratti and Stinson's complete classification through order 11 states that \((v,10)\)-modular Golomb rulers do not exist for \(92\leq v\leq106\). In particular, there is no \((100,10)\)-ruler. The executable artifact independently exhausts the case \(v=100\). Since any simultaneous set must satisfy the product condition, its size is at most 9. The displayed set attains that bound.

## Research packet

### Saved solution and evidence

**Computation 1 (The exact maximum is 9).** A nine-element set meets both Sidon conditions, while the multiplicative condition alone excludes size 10.

The answer is
\[
\max |A|=9.
\]
A witness is
\[
A=\{1,11,47,62,67,70,71,84,96\}.
\]
Its 45 unordered pair sums with repetition are distinct in \(\mathbb F_{101}\), and its 45 corresponding products are also distinct.

For the upper bound, 2 generates \(\mathbb F_{101}^{*}\), which has order 100. Write every element of a product-Sidon set as \(2^b\), with \(b\in\mathbf Z/100\mathbf Z\). Distinct pair products are equivalent to distinct pair sums of the exponents. This is also equivalent to requiring all directed differences \(b_i-b_j\), for \(i\ne j\), to be distinct. Thus a size-10 product-Sidon set would give a \((100,10)\)-modular Golomb ruler.

Buratti and Stinson's complete classification through order 11 states that \((v,10)\)-modular Golomb rulers do not exist for \(92\leq v\leq106\). In particular, there is no \((100,10)\)-ruler. The executable artifact independently exhausts the case \(v=100\). Since any simultaneous set must satisfy the product condition, its size is at most 9. The displayed set attains that bound.

### Background and intake notes

Every rejected extension has a named sum or product collision, so branch exclusions and incumbents are compact reusable evidence.

- Original intake status: Status remains unverified. Additive and multiplicative Sidon sets are studied separately, while this exact simultaneous finite instance may be new.
- Scale a nonempty set so that 1 belongs to A, then branch while maintaining separate 101-bit masks for occupied sums and products. Quotient any remaining inversion symmetry only after checking that it preserves both collision conditions.
- Trap: pairs with repetition include (a,a). Omitting diagonal pairs changes the upper bound and admits invalid witnesses.

- Recorded example: The set {1,11,47,62,67,70,71,84,96} is valid.

### Prior approaches

- **Route 1** (supported): The coupled field problem appears absent from the sources checked, while its decisive ten-point obstruction is already known. [1](#reference-1) [2](#reference-2) [3](#reference-3)

### Runnable artifacts

- **Artifact 1** (executable material): A deterministic C++ program checks both 45-element images and exhausts every normalized ten-mark candidate. [1](#reference-1)

### Computational notes

- Exact replay of the displayed 9-set produced 45 distinct pair sums and 45 distinct pair products. Hence the maximum is at least 9. A 14-set would have 105 unordered pairs with repetition but only 101 possible sums, giving the current bounds 9<=|A|<=13.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `simultaneous-sum-product-sidon-f101`, 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.

## Lean verification

A solution has been submitted and awaits independent review.

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## References

1. <a id="reference-1"></a>Reference record for “simultaneous-sum-product-sidon-f101,” doi.org, checked 2026-08-01. Packet-wide source named in the dataset provenance. https://doi.org/10.26493/1855-3974.2374.9ff
   - Also cited at Marco Buratti and Douglas R. Stinson, New results on modular Golomb rulers, optical orthogonal codes and related structures, Ars Mathematica Contemporanea 20 (2021), 1-27, definition of a modular Golomb ruler and Table 1; independently replayed in sps101-artifact-witness-and-size-ten-exhaustion
   - Also cited at Inline C++17 computation executed on 2026-07-25
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - Supplies the source context used to assemble the packet for “simultaneous sum product sidon f101.”
   - Source named by the research packet.
2. <a id="reference-2"></a>Scholarly publication identified by DOI 10.1112/jlms/s1-16.4.212, checked 2026-08-01. Exact article identified by this DOI. https://doi.org/10.1112/jlms/s1-16.4.212
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - Provides a source or neighboring result used in the packet for “simultaneous sum product sidon f101.”
3. <a id="reference-3"></a>R. C. Bose and S. Chowla, 1962 Exact article identified by this DOI. https://doi.org/10.1007/BF02566968
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - Provides a source or neighboring result used in the packet for “simultaneous sum product sidon f101.”