Prime classification

Classification remains a distinct 56-class, three-way catalogue. Numerisect 0.5.0 adds separate pages for comparing probable-prime tests with rigorous PARI proof modes, running Lucas–Lehmer/Pépin special-form proofs, and independently validating a machine-readable PARI certificate. These additions do not change a classifier timeout or catalogue boundary into a negative classification.

Numerisect evaluates 56 prime classifications through a native PARI/GP program. Python validates the request, starts PARI/GP, parses tagged results, and writes the report; JavaScript presents the result. Neither interface layer performs the classification mathematics.

Using the classifier

1. Start Numerisect with ./run.sh and open http://127.0.0.1:8765/#primes/prime-classify.
2. In Classify a prime, enter a decimal integer or a supported integer expression such as 2^127 - 1.
3. Choose the per-class time budget. The default is two seconds; available choices range from one to ten seconds.
4. Select Classify prime.

For a versioned user installation on Linux, Windows WSL, or macOS, see Installation and versioning.

PARI/GP rigorously tests the input with isprime before classification. A composite input is reported immediately and the 56 membership tests are not run. A prime result is rendered as classification cards and saved as a text file in output/. Its result appears beneath the form with the exact saved path and a download control. The page is in Primality & navigation; for testing a list without classification, use Check a list of integers in the same group (see Prime manipulation).

Result meanings

Result	Meaning
Match	The native test established that the prime belongs to the class.
Not matched	The native test established that the prime does not belong to the class.
Inconclusive	The test exceeded its time budget or a deliberately documented exact-search/catalogue boundary. It is not a negative result.

Each matching or inconclusive item includes an identifier, display name, definition, and, when useful, native-engine detail such as a partner prime, sequence index, exponent, or limiting reason. The summary also reports the number of tested, matched, not-matched, and inconclusive classes.

Classification catalogue

All classifications from the source catalogue are represented exactly once. “Exact” below means that Numerisect uses an algebraic condition, recurrence, rigorous primality test, congruence, or exhaustive test for the supplied input.

#	Identifier	Classification	Native test and scope
1	balanced	Balanced	Exact comparison with the nearest proven prime on each side.
2	chen	Chen	Exact test that p + 2 is prime or has exactly two prime factors with multiplicity.
3	circular	Circular	Rigorously tests every cyclic decimal rotation.
4	cluster	Cluster	Exact prime-difference coverage through p = 100000; larger inputs are inconclusive.
5	cousin	Cousin	Rigorously tests partners at distance four.
6	cuban	Cuban	Exact recognition of the two cubic-difference forms.
7	cullen	Cullen	Exact recognition of n * 2^n + 1.
8	delicate	Digitally delicate	Rigorously tests every one-digit decimal replacement.
9	dihedral	Dihedral	Rigorously tests the valid seven-segment rotations and reflections.
10	double_mersenne	Double Mersenne	Exact recognition of 2^(2^q - 1) - 1 with prime q.
11	emirp	Emirp	Rigorously tests the distinct decimal reversal.
12	even	Even	Exact recognition of the unique even prime.
13	factorial	Factorial	Exact recognition of n! - 1 and n! + 1.
14	fermat	Fermat	Exact recognition of 2^(2^n) + 1.
15	fibonacci	Fibonacci	Exact square-criterion sequence membership.
16	fortunate	Fortunate	Exact primorial search for candidate values through 1000; larger candidates are inconclusive.
17	good	Good	Exact symmetric-neighbor comparisons through p = 1000000; larger inputs are inconclusive.
18	happy	Happy	Exact digit-square orbit with cycle detection.
19	higgs	Higgs	Exact exponent-two sequence construction through p = 100000; larger inputs are inconclusive.
20	left_and_right_truncatable	Left-and-right truncatable	Rigorously tests simultaneous outer decimal truncations.
21	left_truncatable	Left-truncatable	Rigorously tests all left truncations and rejects embedded leading-zero cases.
22	lucas	Lucas	Exact square-criterion sequence membership.
23	mersenne	Mersenne	Exact recognition that p + 1 is a power of two.
24	mills	Mills	Membership in the established Mills-prime sequence; values beyond the established catalogue are inconclusive.
25	minimal	Minimal	Membership in the established minimal-prime catalogue.
26	motzkin	Motzkin	Exact integer-recurrence sequence membership.
27	non_insertable	Non-insertable	Rigorously tests every decimal digit at every insertion position.
28	newman_shanks_williams	Newman–Shanks–Williams	Exact recurrence membership with the required eligible odd index.
29	palindromic	Palindromic	Exact decimal reversal comparison.
30	pell	Pell	Exact integer-recurrence sequence membership.
31	pell_lucas	Pell–Lucas	Exact half-companion Pell recurrence membership.
32	permutable	Permutable	Established non-repunit catalogue plus exact repunit recognition; unresolved range beyond the catalogue is inconclusive.
33	pierpont	Pierpont	Exact recognition of 2^u * 3^v + 1.
34	pillai	Pillai	Exact factorial-residue search through p = 200000; larger inputs are inconclusive.
35	prime_quadruplet	Prime quadruplet	Rigorously tests every constellation position in {q, q+2, q+6, q+8}.
36	prime_triplet	Prime triplet	Rigorously tests all three-prime constellations of diameter six.
37	primorial	Primorial	Exact recognition of one more or less than a prime primorial.
38	proth	Proth	Exact recognition of k * 2^n + 1, where k is odd and k < 2^n.
39	pythagorean	Pythagorean	Exact congruence test p = 1 (mod 4).
40	quartan	Quartan	Exact binary-quadratic-form solution followed by fourth-power verification.
41	repunit	Repunit	Exact recognition of an all-ones decimal representation.
42	right_truncatable	Right-truncatable	Rigorously tests every successive right truncation.
43	safe	Safe	Rigorously tests (p - 1) / 2.
44	sexy	Sexy	Rigorously tests partners at distance six.
45	sophie_germain	Sophie Germain	Rigorously tests 2p + 1.
46	strobogrammatic	Strobogrammatic	Exact 180-degree decimal-display transformation.
47	strong	Strong	Exact comparison with the mean of the nearest proven prime neighbors.
48	superprime	Superprime	Exact prime-index test through p = 10^12; larger inputs are inconclusive.
49	supersingular	Supersingular	Exact membership in the prime divisors of the Monster group's order.
50	twin	Twin	Rigorously tests partners at distance two.
51	wagstaff	Wagstaff	Exact recognition of (2^q + 1) / 3 with odd prime q.
52	wieferich	Wieferich	Exact base-two congruence modulo p^2.
53	williams	Williams	Exact form test for the catalogue's tested bases 3 through 9.
54	wilson	Wilson	Established members and exhaustive non-membership below 2 * 10^13; larger inputs are inconclusive.
55	wolstenholme	Wolstenholme	Established members and exhaustive non-membership below 10^9; larger inputs are inconclusive.
56	woodall	Woodall	Exact recognition of n * 2^n - 1.

API

The endpoint accepts the same restricted integer-expression syntax as the other Numerisect tools. First create the local session cookie as documented in the security model, then run:

curl --cookie numerisect.cookies -X POST http://127.0.0.1:8765/api/primes/classify \
  -H 'Content-Type: application/json' \
  -d '{"expression":"2^127 - 1","per_test_seconds":2}'

The response has this shape:

{
  "number": "170141183460469231731687303715884105727",
  "digits": 39,
  "is_prime": true,
  "classification": "prime",
  "matches": [],
  "inconclusive": [],
  "tested": 56,
  "not_matched": 56,
  "engine": "PARI/GP",
  "note": "...",
  "output_file": "..."
}

The contents of matches and inconclusive depend on the input and budget; the empty arrays above document the response structure rather than the actual classification of the example number.

Time and size behavior

The selected budget applies independently to every guarded classification. PARI/GP's alarm mechanism interrupts a native test that exceeds its budget, and Numerisect records that class as inconclusive. The overall subprocess has a larger safety timeout so all 56 tests normally have time to finish.

Arbitrary-precision arithmetic is provided by PARI/GP and GMP. There is no fixed-precision limit in the native primality and exact algebraic operations, but API expressions are capped at 100,000 characters and evaluated results follow the configured expression-size limit. Time and memory grow with input size. Explicit boundaries in the table protect interactive use where a definition requires an exhaustive search or where mathematical knowledge is catalogue-based.

PARI's candidate-navigation routines may return pseudoprimes for sufficiently large inputs. The classifier's neighbor helper therefore rechecks candidates with rigorous isprime before using them in a proof-oriented result.

Implementation map

File	Responsibility
numerisect/prime_classifier.gp	All classification mathematics, rigorous primality checks, time guards, and tagged native output.
numerisect/primes.py	Catalogue metadata, GP process boundary, tagged-output validation, and response assembly.
numerisect/main.py	Request validation, HTTP endpoint, error translation, and text-report persistence.
numerisect/static/index.html	Classifier form and time-budget controls.
numerisect/static/app.js	Request submission and result rendering.
numerisect/static/styles.css	Match, non-match summary, and inconclusive presentation.
tests/test_primes.py	Native integration and response-contract regression coverage.

When a classification is extended, its identifier must stay synchronized between the GP dispatcher and the Python metadata catalogue. A mathematical limit must be exposed as inconclusive unless non-membership is actually proven.