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

Annotations already declare readOnlyHint, idempotentHint, and non-destructive behavior, so the safety profile is covered. The description adds genuinely new behavior context: the pricing model ($0.001 USDC per successful call via x402 on Solana) and an equivalent HTTPS endpoint, which an agent would not otherwise know.

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

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

Front-loaded with the predicate definition, then examples, then domain, then payment mechanics. Dense but each sentence adds information; only mild cost is that the HTTPS endpoint sentence is a bolted-on alternative transport that slightly breaks flow.

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

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

Definition, examples, domain, cost, and transport are all present, which is ample for a single-integer predicate with no output schema. Remaining gap is the exact response shape (boolean vs. structured object) and how the x402 payment flow is triggered, which an agent calling for the first time might need.

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

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

Schema coverage is 0%, so the description must carry the load. It does specify what n is and restates the 1-20000 domain (mirroring the schema min/max but making the constraint visible in prose), plus examples that show accepted values and their results. It does not clarify the cost of an invalid or out-of-range n, but the core semantics are covered.

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

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

States a specific predicate with a precise mathematical definition (composite, squarefree, p-1 | n-1) plus concrete positive and negative examples. It is clear what the tool computes, but it never distinguishes itself from the sibling 'carmichael', so an agent cannot tell whether that sibling is a listing, a different predicate, or an alias.

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

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

The valid input domain (1 to 20000) is given, which implicitly bounds when the tool can be used, and the examples clarify the semantics. However, there is no explicit guidance on when to choose this over carmichael, carmichael_next, or carmichael_count, leaving the sibling routing to inference.

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