SageMath MCP Server

Number Theory Operation

number_theory_operation

Idempotent

Perform primality testing, integer factorization, next prime, gcd, and lcm calculations with SageMath's reliable number theory engine.

Instructions

Number theory: primality testing, integer factorisation, the next prime above n, gcd and lcm. Prefer this over evaluate_sage for any of these.

Input Schema

TableJSON Schema

Name	Required	Description	Default
`a`	Yes	Primary integer. Pass values above 2^53 as a decimal STRING: JSON numbers are IEEE doubles in JavaScript-based clients, so 10^30 arrives as 1000000000000000019884624838656 and the answer is silently wrong.
`b`	No	Second integer, required for gcd and lcm. Same string rule.
`session`	No	Workspace to use, as a name or a portable handle. Workspaces have independent variables. A name is scoped to this MCP session; a handle returned by start_sage_session (workspace_token) reaches the same workspace across reconnects and is a bearer credential -- keep it secret. Omit for 'default'.	default
`operation`	Yes	Operation: 'is_prime', 'factor_integer', 'next_prime', 'gcd', 'lcm'

Output Schema

TableJSON Schema

Name	Required	Description	Default
No arguments

Schema Changelog

Changes observed during successful MCP inspections.

Changed1 schema field changedv0.7.0Sep 12, 2026

changedInput schema / properties / session / description

Previous value: -"Named workspace to use. Workspaces have independent variables; omit for 'default'."New value: +"Workspace to use, as a name or a portable handle. Workspaces have independent variables. A name is scoped to this MCP session; a handle returned by start_sage_session (workspace_token) reaches the same workspace across reconnects and is a bearer credential -- keep it secret. Omit for 'default'."

Changed6 schema fields changedv0.5.0Aug 14, 2026

addedInput schema / properties / a / anyOf

Added value: +[
+  {
+    "type": "integer"
+  },
+  {
+    "type": "string"
+  }
+]

changedInput schema / properties / a / description

Previous value: -"Primary integer argument"New value: +"Primary integer. Pass values above 2^53 as a decimal STRING: JSON numbers are IEEE doubles in JavaScript-based clients, so 10^30 arrives as 1000000000000000019884624838656 and the answer is silently wrong."

removedInput schema / properties / a / type
```
Removed value: -"integer"
```

changedInput schema / properties / b / anyOf

Previous value: -[
-  {
-    "type": "integer"
-  },
-  {
-    "type": "null"
-  }
-]New value: +[
+  {
+    "type": "integer"
+  },
+  {
+    "type": "string"
+  },
+  {
+    "type": "null"
+  }
+]

changedInput schema / properties / b / description

Previous value: -"Second integer (required for gcd, lcm)"New value: +"Second integer, required for gcd and lcm. Same string rule."

addedInput schema / properties / session

Added value: +{
+  "default": "default",
+  "description": "Named workspace to use. Workspaces have independent variables; omit for 'default'.",
+  "type": "string"
+}

First observedv0.3.1Jul 31, 2026

TDQS

A4.2/5.0

Behavior3/5

Does the description disclose side effects, auth requirements, rate limits, or destructive behavior?

Annotations already declare idempotentHint=true and destructiveHint=false, so the safety profile is covered. The description adds no behavioral context beyond listing operations, which is acceptable given the annotations. No contradiction exists, but it does not elaborate on return format or error behavior, which the output schema presumably covers.

Agents need to know what a tool does to the world before calling it. Descriptions should go beyond structured annotations to explain consequences.

Conciseness5/5

Is the description appropriately sized, front-loaded, and free of redundancy?

The description is a single, front-loaded sentence that states the purpose and then gives the routing directive. There is zero redundancy or filler; every word earns its place.

Shorter descriptions cost fewer tokens and are easier for agents to parse. Every sentence should earn its place.

Completeness4/5

Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?

With a rich output schema present, the description does not need to explain return values. It covers the full set of operations and the preferred alternative. The only minor gap is that it doesn't spell out which operations require the b parameter, but the schema's description for b explicitly states 'required for gcd and lcm,' so this is adequately covered elsewhere.

Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.

Parameters3/5

Does the description clarify parameter syntax, constraints, interactions, or defaults beyond what the schema provides?

Schema description coverage is 100%, so every parameter (a, b, session, operation) is already fully documented in the input schema, including the string rule for large integers and session semantics. The description adds no parameter-specific detail, which is fine because the schema carries the burden.

Input schemas describe structure but not intent. Descriptions should explain non-obvious parameter relationships and valid value ranges.

Purpose5/5

Does the description clearly state what the tool does and how it differs from similar tools?

The description explicitly states the tool's purpose: 'Number theory: primality testing, integer factorisation, the next prime above n, gcd and lcm.' This is a specific verb-resource pairing that clearly distinguishes it from the generic evaluate_sage sibling by naming the exact operations covered.

Agents choose between tools based on descriptions. A clear purpose with a specific verb and resource helps agents select the right tool.

Usage Guidelines5/5

Does the description explain when to use this tool, when not to, or what alternatives exist?

It gives explicit routing guidance: 'Prefer this over evaluate_sage for any of these.' This names the alternative tool and specifies the condition (any of the listed number theory operations) for choosing this tool, leaving no ambiguity for the agent.

Agents often have multiple tools that could apply. Explicit usage guidance like "use X instead of Y when Z" prevents misuse.