{ "openapi": "3.1.1", "info": { "title": "Number-Theory Lab API", "version": "1.0.0", "summary": "Primality, factorisation, gcd, modular arithmetic, CRT, φ, π(n), Collatz, continued fractions.", "description": "Every endpoint performs the same calculation as the human page and returns inputs, result, units, assumptions, sources, warnings and computedAt. MCP endpoint: /mcp." }, "servers": [ { "url": "https://numbertheorylab.usefulweb.site" } ], "paths": { "/api/v1/calculate": { "post": { "operationId": "calculate", "summary": "Primality, factorisation, gcd, modular arithmetic, CRT, φ, π(n), Collatz, continued fractions.", "description": "Big-integer number theory with the working shown: Miller–Rabin primality (deterministic up to 3.3·10^24, otherwise with a stated number of rounds), factorisation by trial division and Pollard rho, gcd/lcm with the extended Euclidean table, modular inverse and exponentiation, the Chinese remainder theorem, Euler φ and divisors, prime counting up to 10 million, Collatz sequences and continued fractions.\n\nMethodology:\n- Primality: trial division by primes below 100, then Miller–Rabin. Below 3,317,044,064,679,887,385,961,981 the first 13 primes as bases form a proof (Sorenson & Webster 2015), which covers every 64-bit number. Above it, the stated number of rounds uses bases derived from SHA-256 of n (reproducible); a composite passes each round with probability at most 1/4.\n- Factorisation: trial division up to 10,000, then Pollard–Brent rho with a fixed iteration budget; factors are checked with Miller–Rabin. If the budget runs out the remaining cofactor is reported as unfactored, never guessed.\n- φ(n) = ∏ p^(e−1)(p − 1), σ(n) = ∏ (p^(e+1) − 1)/(p − 1), d(n) = ∏ (e + 1). Carmichael numbers are detected with Korselt's criterion (square-free, composite, p − 1 | n − 1 for every prime factor).\n- gcd via the Euclidean algorithm with Bézout coefficients; inverse exists iff gcd(a, m) = 1. CRT merges congruences pairwise and also handles non-coprime moduli (or reports that none exists). π(n) uses an odd-only sieve of Eratosthenes.", "tags": [ "math" ], "x-mcp-tool": "number_theory_lab", "requestBody": { "required": true, "content": { "application/json": { "schema": { "$ref": "#/components/schemas/NumberTheoryLabInput" }, "examples": { "example1": { "summary": "561, the smallest Carmichael number", "value": { "operation": "analyze", "a": "561" } }, "example2": { "summary": "Factor 2^64 + 1", "value": { "operation": "analyze", "a": "2^64+1" } }, "example3": { "summary": "Extended Euclid of 240 and 46", "value": { "operation": "gcd", "a": "240", "b": "46" } }, "example4": { "summary": "CRT: x ≡ 2 (3), 3 (5), 2 (7)", "value": { "operation": "crt", "congruences": "2 3\n3 5\n2 7" } }, "example5": { "summary": "π(10^6)", "value": { "operation": "primecount", "a": "1000000" } }, "example6": { "summary": "Continued fraction of √7", "value": { "operation": "cf", "a": "7" } } } } } }, "responses": { "200": { "description": "Calculation result envelope.", "content": { "application/json": { "schema": { "$ref": "#/components/schemas/NumberTheoryLabResult" } } } }, "400": { "description": "Malformed JSON body.", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "422": { "description": "Invalid input (see errors[]).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "429": { "description": "Rate limit exceeded (see Retry-After).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } } } } }, "/api/v1/tool": { "get": { "operationId": "getToolDefinition", "summary": "Inputs, output schema and examples for this tool", "responses": { "200": { "description": "Tool definition.", "content": { "application/json": { "schema": { "type": "object" } } } } } } } }, "components": { "schemas": { "Problem": { "type": "object", "description": "RFC 9457 problem details. 422 responses include per-field `errors`.", "required": [ "type", "title", "status", "detail" ], "properties": { "type": { "type": "string" }, "title": { "type": "string" }, "status": { "type": "integer" }, "detail": { "type": "string" }, "code": { "type": "string" }, "errors": { "type": "array", "items": { "type": "object", "required": [ "field", "message" ], "properties": { "field": { "type": "string" }, "message": { "type": "string" } } } } } }, "NumberTheoryLabInput": { "type": "object", "additionalProperties": false, "properties": { "operation": { "title": "Operation", "type": "string", "enum": [ "analyze", "gcd", "modpow", "modinv", "crt", "primecount", "collatz", "cf" ], "default": "analyze" }, "a": { "title": "n or a", "description": "Integer: decimal, 0x hex, 0b binary, or a power such as 2^127-1.", "type": "string", "maxLength": 1300, "default": "561" }, "b": { "title": "b (second number / exponent / denominator)", "type": "string", "maxLength": 1300 }, "m": { "title": "Modulus m", "type": "string", "maxLength": 1300 }, "congruences": { "title": "Congruences (one per line: remainder modulus)", "description": "For the CRT: each line \"a m\" means x ≡ a (mod m). Moduli need not be coprime.", "type": "string", "maxLength": 4000 }, "rounds": { "title": "Miller–Rabin rounds above 3.3·10^24", "description": "Error probability at most 4^−rounds for numbers beyond the deterministic range.", "type": "integer", "minimum": 1, "maximum": 64, "default": 40 } }, "required": [] }, "NumberTheoryLabResult": { "type": "object", "additionalProperties": false, "required": [ "schemaVersion", "tool", "computedAt", "inputs", "summary", "result", "units", "assumptions", "sources", "warnings", "presentation", "links" ], "properties": { "schemaVersion": { "type": "string", "const": "1.0" }, "tool": { "type": "object", "additionalProperties": false, "required": [ "id", "name", "version" ], "properties": { "id": { "type": "string" }, "name": { "type": "string" }, "version": { "type": "string" } } }, "computedAt": { "type": "string", "format": "date-time" }, "inputs": { "type": "object", "description": "Normalised inputs actually used, including applied defaults." }, "summary": { "type": "string" }, "result": { "type": "object", "required": [ "operation", "answer" ], "properties": { "operation": { "type": "string" }, "answer": { "type": "string", "description": "Main result as text (big integers as decimal strings)." }, "n": { "type": "string" }, "isPrime": { "type": "boolean" }, "primalityMethod": { "type": "string" }, "errorBound": { "type": "number" }, "basesTested": { "type": "array", "items": { "type": "string" } }, "factors": { "type": "array", "items": { "type": "object", "required": [ "p", "e" ], "properties": { "p": { "type": "string" }, "e": { "type": "integer" } } } }, "factorisationComplete": { "type": "boolean" }, "phi": { "type": [ "string", "null" ] }, "divisorCount": { "type": [ "string", "null" ] }, "divisorSum": { "type": [ "string", "null" ] }, "divisors": { "type": [ "array", "null" ], "items": { "type": "string" } }, "carmichael": { "type": "boolean" }, "gcd": { "type": "string" }, "lcm": { "type": "string" }, "bezout": { "type": "object", "properties": { "x": { "type": "string" }, "y": { "type": "string" } } }, "value": { "type": [ "string", "null" ] }, "modulus": { "type": "string" }, "count": { "type": "integer" }, "steps": { "type": "integer" }, "peak": { "type": "string" }, "terms": { "type": "array", "items": { "type": "string" } }, "period": { "type": "array", "items": { "type": "string" } }, "convergents": { "type": "array", "items": { "type": "string" } } } }, "units": { "type": "object", "additionalProperties": { "type": "string" } }, "assumptions": { "type": "array", "items": { "type": "string" } }, "sources": { "type": "array", "items": { "type": "object", "additionalProperties": false, "required": [ "title" ], "properties": { "title": { "type": "string" }, "url": { "type": "string" }, "publisher": { "type": "string" }, "retrievedAt": { "type": "string" }, "note": { "type": "string" } } } }, "warnings": { "type": "array", "items": { "type": "string" } }, "presentation": { "type": "array", "items": { "type": "object" }, "description": "Display blocks used by the human page (same data)." }, "links": { "type": "object", "additionalProperties": false, "required": [ "human", "api", "openapi", "methodology" ], "properties": { "human": { "type": "string", "format": "uri" }, "api": { "type": "string", "format": "uri" }, "openapi": { "type": "string", "format": "uri" }, "methodology": { "type": "string", "format": "uri" } } } } } } } }