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

Annotations already declare a safe, idempotent, read-only, non-destructive operation, so the added value here is the input domain cap (1 to 1,000,000), the priced-per-successful-call model ($0.001 USDC via x402 on Solana), and a documented HTTPS fallback endpoint — details an agent cannot get from the schema. It does not explicitly state the return value shape, though the 'Whether' framing implies a boolean.

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, followed by examples and constraints in a compact sequence of short sentences. The example enumeration (21, 33, 77 / 9, 15, 25, 35, 49) is slightly longer than needed but genuinely disambiguates the definition.

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?

For a one-parameter predicate with no output schema, the critical pieces are present: the definition, range, cost, and alternate endpoint. What is missing is an explicit statement of the returned boolean and any routing guidance versus the four sibling Blum tools, which matters given how crowded this namespace is.

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 burden, and it does partially: 'n' is identified as the integer under test and 'n runs from 1 to 1000000' restates the schema's own minimum/maximum rather than adding new meaning. The mathematical definition plus worked examples clarify what n represents semantically, but nothing is said about invalid input handling or non-integer values.

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?

The description states a precise mathematical predicate — whether n is the product of two distinct primes both congruent to 3 mod 4 — with concrete positive and negative examples, so an agent knows exactly what the tool computes. It does not name the closely related siblings (blum_count, blum_next, blum_factors, blum_sum), so differentiation is left implicit.

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?

It states the valid input range (1 to 1000000) and pricing, but gives no guidance on when to choose this predicate over blum_count, blum_next, blum_factors, or blum_sum, nor any exclusion conditions. An agent must infer that this is the single-value boolean test among a family of Blum tools.

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