HTTP API reference

Every route below is generated from the running application, so this page cannot drift from the code. Interactive OpenAPI documentation is also served at /api/docs while Numerisect is running.

Authentication

All /api/ routes except /api/session require the per-launch session token, supplied either as the X-Numerisect-Token header or the numerisect_session cookie. Obtain it from GET /api/session. The token changes every time the process starts.

curl --cookie-jar jar http://127.0.0.1:8765/api/session
curl --cookie jar --header 'Content-Type: application/json' \
  --data '{"expression":"32416190071","mode":"proven"}' \
  http://127.0.0.1:8765/api/primes/check

To avoid tokens entirely, use numerisect api, which drives the same application in-process. See the command-line reference.

Conventions

• Invalid input returns 422 with a detail message naming the problem.
• A missing engine returns 503.
• A missing, stale, or cross-site local session and a refused optional network action return 403, with the cause in detail. Since tokens are per process, a client must repeat GET /api/session after every server restart.
• Operations that produce a result save a report and return its exact output/<filename> path.
• Searches that stop at a bound set a truncation flag and give a continuation point.

Routes (228)

Engine adapters

Method	Path	Purpose
GET	/api/adapters	List built-in and user-declared engine adapters.

Algebra laboratory

Method	Path	Purpose
POST	/api/algebra/chebotarev	Chebotarev density and Frobenius patterns. Factor a monic irreducible polynomial modulo every unramified prime below a bound and compare the observed cycle-type frequencies with the conjugacy-class densities of its Galois group.
POST	/api/algebra/congruence	Solve P(x) ≡ 0 (mod m). Factor the modulus, solve modulo each prime, lift with Hensel's lemma including singular roots, and recombine every branch with the Chinese remainder theorem.
POST	/api/algebra/cornacchia	Solve x² + dy² = n by Cornacchia. Solve r² ≡ −d for every square divisor of n, run the Euclidean descent with a full step trace, and cross-check the complete solution set against PARI's qfbsolve.
POST	/api/algebra/discrete-log	Solve gˣ ≡ h (mod n) by a chosen method. Run baby-step/giant-step, Pohlig–Hellman, Pollard rho, or PARI's native znlog under an explicit budget of group operations.
POST	/api/algebra/divisor-lattice	Enumerate every divisor and factor pair. List each divisor with its cofactor and Ω value, and build the covering relation of the divisor lattice when the divisor count stays inside the configured cap.
POST	/api/algebra/factor-stages	Factor a polynomial over F_p one classical stage at a time: square-free decomposition, distinct-degree factorization of each part, then equal-degree splitting, with the reconstruction verified by the engine.
POST	/api/algebra/finite-field	Compute in 𝔽ₚ and 𝔽_{pᵐ}. Build the field from a supplied irreducible reduction polynomial or from PARI's ffinit, then add, subtract, multiply, divide, exponentiate, and report orders, minimal polynomials, and Frobenius images.
POST	/api/algebra/number-field	Split rational primes in a number field. Build the maximal order of a monic irreducible polynomial and report splitting, inertia, ramification, ideal norms, an element's ideal factorization, and the class group.
POST	/api/algebra/quadratic-ring	Norms, units, and primes in ℚ(√d). Compute the norm and trace of a + bω, decide whether it is a unit or a prime of the ring, report the fundamental unit, roots of unity, and class group, and split a rational prime with explicit generators.
POST	/api/algebra/reciprocity	Quadratic reciprocity and the Jacobi symbol (a/n). Reduce (a/n) step by step with reciprocity, the supplementary law for 2, and modular reduction, then cross-check the sign against the native Kronecker symbol.
POST	/api/algebra/record-numbers	Highly composite and colossally abundant numbers. Test n for record τ and record σ(n)/n, decide the superior-highly-composite and colossally-abundant properties from their exact ε-intervals, and list all four record sequences.
POST	/api/algebra/smoothness	Bound the prime factors of n. Report the exact least prime factor, largest prime factor, largest prime power, and the B-smooth, B-powersmooth, and R-rough verdicts with the smooth and rough parts.
POST	/api/algebra/sociable	Aliquot cycles: amicable pairs and sociable chains. Iterate s(n) = σ(n) − n over an interval and report every closed cycle.
POST	/api/algebra/weird-numbers	Abundant, perfect, multiperfect & weird. Classify n by its proper-divisor sum and decide semiperfection with an exact subset-sum over the proper divisors, returning a witness subset whenever one exists.

Batch import

Method	Path	Purpose
POST	/api/batch/import	Expand a TXT/CSV/JSON document of integers, ranges, expressions, and families.

Result cache

Method	Path	Purpose
DELETE	/api/cache	Discard cached results, optionally for one operation path only.
GET	/api/cache	Report result-cache size and hit counts.

Capabilities

Method	Path	Purpose
GET	/api/capabilities	Report which engines are installed and what the build supports.

Optional catalogues

Method	Path	Purpose
GET	/api/catalogues	Show local catalogue files and the current network permission state.
POST	/api/catalogues/factors	Look up claimed factors locally, or remotely when explicitly permitted.
POST	/api/catalogues/oeis	Search OEIS for a sequence. Disabled unless explicitly permitted.

Prime counting

Method	Path	Purpose
POST	/api/counting/algorithm-comparison	Count π(x) with distinct algorithms. primecount supplies six mathematically distinct algorithms in one codebase; PARI/GP primepi adds an independent engine implementation.
POST	/api/counting/nth-prime-inverses	Compare Li⁻¹(n) and R⁻¹(n) with the n-th prime. Both inverse approximations estimate the n-th prime: Li⁻¹ inverts the Eulerian logarithmic integral and R⁻¹ inverts the Riemann R function.
POST	/api/counting/phi	Evaluate Legendre's phi(x, a). φ(x, a) counts the integers in [1, x] divisible by none of the first a primes — the partial sieve inside Legendre's, Meissel's and Lehmer's formulas.

Diagnostics

Method	Path	Purpose
POST	/api/diagnostics	Collect an environment and engine diagnostic bundle for bug reports.

Distributed CADO-NFS

Method	Path	Purpose
POST	/api/distributed/factor	Queue a factorization whose sieving is distributed by CADO-NFS.
POST	/api/distributed/preview	Validate a distributed configuration and report its exposure without running.
GET	/api/distributed/trust-model	Describe how distributed CADO authenticates clients, and what it does not.

Analytic distribution

Method	Path	Purpose
POST	/api/distribution/approximation-error	Chart π(x) approximation error. Plot the absolute and relative error of x/log x, li(x), and R(x) against exact π(x) over a log-spaced grid.
POST	/api/distribution/bateman-horn	Bateman–Horn prediction for polynomials. Prove each polynomial irreducible, count ω(p) with polrootsmod, build the Bateman–Horn constant, and compare the prediction with an exact count over a bounded range.
POST	/api/distribution/density-surface	Map density over position and residue. Bin every prime in a range by block position and reduced residue class modulo q, then draw the normalised density as a heatmap.
POST	/api/distribution/maximal-gaps	Search for maximal gaps.
POST	/api/distribution/nth-prime-bounds	Check Rosser–Schoenfeld and Dusart bounds. Evaluate six published upper and lower bounds for pₙ at high precision and compare each with the exact n-th prime from primecount.
POST	/api/distribution/pnt-convergence	Track π(x)/(x/log x) and π(x)/li(x). Follow both prime-number-theorem ratios and the normalised error term across a log-spaced grid, flagging every sign change of π(x) − li(x) that the grid can see.
POST	/api/distribution/prime-race	Prime races and Chebyshev bias. Count primes in every reduced residue class modulo q at a series of checkpoints, report the leader at each checkpoint, and normalise the bias by log x/√x.
POST	/api/distribution/progressions	Compare π(x; q, a) with li(x)/φ(q). Count primes in every reduced class modulo q and report the observed deviation from the expected li(x)/φ(q), in absolute, relative, and √x-normalised form.
POST	/api/distribution/short-interval	Maier matrix short-interval experiment. Count the primes in each row interval [qk + 1, qk + y] and compare with the naive expectation y/log(qk).
POST	/api/distribution/singular-series	Hardy–Littlewood singular series 𝔖. Check admissibility and evaluate the singular series as a truncated Euler product with a rigorous relative tail bound and an explicit enclosing interval.
POST	/api/distribution/tuple-prediction	Count twin primes and constellations. Compare the Hardy–Littlewood prediction 𝔖·∫ dt/logᵏ t with an exact count.

OpenAPI documentation

Method	Path	Purpose
GET	/api/docs	Serve the interactive OpenAPI documentation for this API.

Exports

Method	Path	Purpose
GET	/api/exports/jobs	Export job search results in any supported interchange format.
GET	/api/exports/jobs/{job_id}	Export one factorization job in any supported interchange format.
GET	/api/exports/reports/{filename}	Re-container a saved text report; the engine-produced body is unchanged.

Expert factorization laboratory

Method	Path	Purpose
GET	/api/factor/cado-stages	List CADO-NFS's own workflow stages in order, with the task parameter that halts each one.
POST	/api/factor-lab/certificates	Generate and independently verify a primality certificate per prime factor.
POST	/api/factor-lab/special-form	Detect special algebraic forms, algebraic factors, and SNFS suitability.
POST	/api/factor-lab/classic	Factor with CFRAC, Lehman's deterministic method or Hart's one-line factorization in the compiled helper, reporting the split with each part's primality decided by PARI/GP.
POST	/api/factor-lab/coppersmith	Coppersmith's method: recover a divisor of N from partial knowledge of it, reporting the proven validity window and each root's divisor with its primality verdict.
POST	/api/factor-lab/mersenne-factors	Trial-factor M_p = 2^p − 1 without constructing it: q = 2kp + 1 for prime p, or q = 2kd + 1 over every order divisor d > 1 of an odd composite p.
POST	/api/factor-lab/mersenne-hunt	Run bounded Mersenne trial factoring, GMP-ECM P−1/P+1/ECM stages, and PARI/GP multiplicity, divisibility, cofactor, and primality reconciliation.
GET	/api/factor-lab/sievers	Report the GGNFS lattice sievers found, whether each runs on this CPU, and which directory YAFU will be given for number field sieve work.
GET	/api/factor-lab/rsa-catalogue	List the RSA Factoring Challenge numbers with their sizes and published status.
POST	/api/factor-lab/rsa-challenge	Identify an RSA challenge number, re-verify its published factors in PARI/GP, and estimate the number field sieve effort on this machine.
POST	/api/factor-lab/squfof	Factor with Shanks' square forms factorization (numerisect-squfof, C/GMP).
POST	/api/factor-lab/strategy	Recommend an engine and estimate the expected remaining factor size.
POST	/api/factor-lab/trace	Produce a bounded, educational step trace of a factoring algorithm.
POST	/api/factor-lab/tune	Measure this machine's SIQS/NFS crossover with YAFU's own tune.

Quadratic forms and continued fractions

Method	Path	Purpose
POST	/api/forms/class-group	Class number, group structure, generators and regulator for a discriminant.
POST	/api/forms/compose	Gaussian composition and powers, with and without reduction; principality and order.
POST	/api/forms/continued-fraction	Continued fraction of a rational or a quadratic irrational; partial quotients, convergents, preperiod and period. Values wider than preview_digits are abbreviated in the response and written in full to export_file.
POST	/api/forms/pell	Pell's equation x² − dy² = 1, solved from the fundamental unit and list further solutions. Values wider than digit_limit are abbreviated in the response and written in full to export_file.
POST	/api/forms/prime-form	qfbprimeform for each requested prime.
POST	/api/forms/reduce	Reduce a binary quadratic form, trace the steps, give the SL(2, ℤ) matrix.
POST	/api/forms/reduced-forms	Enumerate the reduced forms of a discriminant; flag ambiguous and square forms.
POST	/api/forms/represent	Represent an integer by a form with qfbsolve.

Performance history

Method	Path	Purpose
GET	/api/history/performance	Aggregate completed-job runtimes by engine and decimal-digit bucket.

Factorization jobs

Method	Path	Purpose
GET	/api/jobs	List jobs, optionally filtered and sorted (roadmap item 135).
POST	/api/jobs	Queue a factorization job on a chosen backend and return its identifier.
POST	/api/jobs/batch	Queue multiple factorizations. Enter one integer expression per line or import a TXT, CSV, or JSON array.
POST	/api/jobs/batch-export	Export several factorization jobs at once in a chosen interchange format.
POST	/api/jobs/reorder	Reorder queued jobs; the supplied order becomes the dispatch order.
GET	/api/jobs/facets	Count the jobs a search matches, grouped by status, engine and digit band.
GET	/api/jobs/{job_id}	Fetch one factorization job's status, phase and result.
POST	/api/jobs/{job_id}/cancel	Cancel a running job by signalling its engine process group.
POST	/api/jobs/{job_id}/cado-stage	Continue a staged CADO-NFS job up to a later stage of its workflow.
POST	/api/jobs/{job_id}/certificates	Certify every prime factor of a completed factorization (roadmap item 13).
POST	/api/jobs/{job_id}/continue-cofactor	Start a linked child job on a composite cofactor the parent could not split.
GET	/api/jobs/{job_id}/export	Download one factorization job in a chosen interchange format.
GET	/api/jobs/{job_id}/log	Stream a job's engine log with absolute paths redacted.
POST	/api/jobs/{job_id}/pause	Suspend a running job's process group with SIGSTOP.
POST	/api/jobs/{job_id}/priority	Change one job's scheduling priority; higher runs sooner.
POST	/api/jobs/{job_id}/resume	Continue a paused job's process group with SIGCONT.
GET	/api/jobs/{job_id}/tree	Arrange a job's factors into the cofactor chain PARI/GP computes, in discovery order.
POST	/api/jobs/{job_id}/resume-paused	Continue a paused job's process group with SIGCONT.

Number theory

Method	Path	Purpose
POST	/api/number-theory/aliquot	Follow an aliquot sequence. Iterate proper-divisor sums with native factorization, exact termination detection, and repeated-value cycle detection.
POST	/api/number-theory/arithmetic-functions	Analyze divisors, kernels & representations. Compute σₖ, Jordan Jₖ, Dedekind ψ, Liouville, von Mangoldt, smoothness, squarefree structure, and quadratic representations.
POST	/api/number-theory/crt	Chinese remainder theorem for a congruence system. Solve compatible systems even when moduli are not pairwise coprime.
POST	/api/number-theory/cunningham-chain	Verify a prime chain. Follow pₖ₊₁ = 2pₖ + 1 or 2pₖ − 1, proving every term and stopping explicitly at the first composite.
POST	/api/number-theory/cyclotomic	Construct Φₙ(x) and factor modulo p. Build the exact cyclotomic polynomial and inspect its irreducible factorization over a selected finite field.
POST	/api/number-theory/discrete-log	Solve gˣ ≡ a (mod n). Use PARI's native algorithm selection with the exact order of the generated subgroup.
POST	/api/number-theory/divisor-classification	Classify an integer by its divisors. Determine deficiency, abundance, perfection, almost-perfection, multiperfection, and amicable pairing exactly.
POST	/api/number-theory/eisenstein	Eisenstein primes a + bω. Apply exact norm and rational-axis criteria in the triangular integer lattice ℤ[ω].
POST	/api/number-theory/factor-strategy	Analyze before factoring. Run native perfect-power, probable-prime, special-form, and small-factor checks before selecting YAFU or CADO-NFS.
POST	/api/number-theory/hensel-roots	Lift polynomial roots modulo pᵏ. Enter integer coefficients from constant term upward and compute exact residues from PARI's p-adic roots.
POST	/api/number-theory/modular-roots	Solve xᵏ ≡ a (mod p). Enumerate all roots in a rigorously verified prime field using PARI finite-field arithmetic.
POST	/api/number-theory/ntt-primes	Generate NTT primes. Find proven primes p = k·2ᵐ + 1 of an exact bit length for native number-theoretic transforms.
POST	/api/number-theory/order-distribution	Distribute multiplicative orders. Enumerate every unit modulo n and count elements of each exact multiplicative order.
POST	/api/number-theory/perfect-power	Detect a perfect power. Find the maximal exponent k and exact base a such that n = aᵏ.
POST	/api/number-theory/polynomial	Factor over ℚ and 𝔽ₚ. Enter integer coefficients from constant term upward; PARI factors the polynomial in both domains and finds finite-field roots.
POST	/api/number-theory/power-residues	Count kth-power residues. Evaluate the complete map x ↦ xᵏ over 𝔽ₚ and count every residue's preimages.
POST	/api/number-theory/primality-lab	Compare primality tests. Compare Fermat, Euler–Jacobi, strong Miller–Rabin, BPSW, and a selected rigorous PARI proof on one input.
POST	/api/number-theory/prime-approximations	Compare π(x) approximations. Compare exact parallel primecount output with x/log(x), Li(x), and Riemann R(x), including signed and relative errors.
POST	/api/number-theory/quadratic-decomposition	Decompose a prime in ℚ(√d). Construct the maximal quadratic order and compute splitting, inertia, ramification, and prime-ideal norms.
POST	/api/number-theory/special-form-test	Test Mersenne or Fermat numbers. Use the necessary-and-sufficient Lucas–Lehmer or Pépin criterion without general factorization.
POST	/api/number-theory/special-prime-family	Search structured prime families. Construct candidates in eight classical families and retain only values rigorously proven prime by PARI/GP.
POST	/api/number-theory/summatory-functions	Calculate M(x), L(x), θ(x), ψ(x). Evaluate Mertens, summatory Liouville, and both Chebyshev functions natively over a bounded exact range.
POST	/api/number-theory/symbols	Legendre, Jacobi & Kronecker. Evaluate all applicable quadratic character symbols exactly and distinguish their domains.
POST	/api/number-theory/tonelli-shanks	Trace Tonelli–Shanks. Compute roots of x² ≡ a (mod p), display the native algorithm state, and verify each result.
POST	/api/number-theory/unit-group	Analyze (ℤ/nℤ)×. Compute invariant factors, component generators, cyclicity, and bounded primitive-root enumeration.
POST	/api/number-theory/valuation	Calculate vₚ(n). Decompose a nonzero arbitrary-size integer into its exact p-power and coprime unit part.

Report download

Method	Path	Purpose
GET	/api/outputs/{filename}	Download a saved report from the output directory.

/api/primality-lab

Method	Path	Purpose
POST	/api/primality-lab/bitwin-chains	Search bi-twin chains. Find maximal chains whose members n·2ⁱ − 1 and n·2ⁱ + 1 are all proven prime.
POST	/api/primality-lab/carmichael	Carmichael numbers and Korselt's criterion. Factor n, test squarefreeness and (p − 1) | (n − 1) for every prime factor, and derive λ(n) and the exact Fermat-liar count.
POST	/api/primality-lab/chernick	Generate Chernick Carmichael numbers. Search k with 6k + 1, 12k + 1 and 18k + 1 simultaneously prime, and re-verify Korselt's criterion for every product.
POST	/api/primality-lab/compare	Compare eight primality tests. Run Fermat, Solovay–Strassen, Miller–Rabin, Lucas, strong Lucas, Frobenius, Baillie–PSW, APR-CL, and ECPP on one integer and keep probable results separate from proofs.
POST	/api/primality-lab/constrained-prime	Generate a proven prime to specification. Construct an exact-bit-length prime with an optional residue condition, safe-prime or Gordon strong-prime structure, and a PARI certificate.
POST	/api/primality-lab/covering-set	Verify a Sierpiński/Riesel covering set. Check exactly that every exponent class modulo the period contributes a fixed prime divisor, which proves k is a Sierpiński or Riesel number.
POST	/api/primality-lab/deterministic-witnesses	Deterministic Miller–Rabin witness sets. Use the published sufficient base sets for bounded ranges, showing every squaring chain and the exact bound that makes the result a proof.
POST	/api/primality-lab/ecpp-steps	Expand an ECPP certificate. Show every Atkin–Morain descent step, its elliptic curve, point, group order, and discriminant, then re-validate the whole certificate.
POST	/api/primality-lab/lucas-lehmer-steps	Show the Lucas–Lehmer iteration. Materialize the residues s₀ = 4, sᵢ = sᵢ₋₁² − 2 modulo Mₚ that decide whether 2ᵖ − 1 is prime.
POST	/api/primality-lab/lucas-sequence	Run Lucas, Frobenius, and Morrison N+1 tests. Select (P, Q) by Selfridge's method A or by hand, then apply the Lucas, strong Lucas, Frobenius, Lucas–Lehmer–Riesel, and Morrison N + 1 criteria.
POST	/api/primality-lab/pocklington	Pocklington N − 1 primality proof. Factor N − 1 far enough for the Pocklington bound, find a witness for every prime divisor of the factored part, and emit a machine-readable certificate.
POST	/api/primality-lab/pratt	Build a recursive Pratt certificate. Certify n from a primitive root and the full factorization of n − 1, then certify every factor recursively down to 2.
POST	/api/primality-lab/prime-ladder	Prime ladders, one digit at a time. Find a shortest path between two equal-width primes in which every intermediate value is a proven prime.
POST	/api/primality-lab/proth	Proth primes k·bⁿ + 1. Apply Proth's necessary-and-sufficient criterion for base 2, or the generalized N − 1 criterion for any other base, and cross-check against PARI isprime.
POST	/api/primality-lab/proth-search	Search k·bⁿ + 1 over an exponent range. Filter candidates with BPSW natively, then settle each survivor with the Proth criterion or PARI isprime, counting expired budgets as inconclusive.
POST	/api/primality-lab/repunit	Generalized repunits (bⁿ − 1)/(b − 1). Skip composite exponents natively, filter with BPSW, and prove every reported repunit prime with PARI isprime.
POST	/api/primality-lab/sierpinski	Sierpiński and Riesel numbers k·2ⁿ ± 1. Find the first proven prime that removes k from the Sierpiński or Riesel candidates; an exhausted bound stays inconclusive.
POST	/api/primality-lab/taxonomy	Classify pseudoprime families. Separate Fermat, Euler–Jacobi, strong, Lucas, strong Lucas, Frobenius, and Baillie–PSW behaviour for one composite and the bases that fool it.
POST	/api/primality-lab/verify-certificate	Verify a Pocklington or Pratt certificate. Re-derive every condition of a certificate from the certificate alone, without consulting the run that produced it.

Prime tools

Method	Path	Purpose
POST	/api/primes/absolute	Find circular primes. Find primes whose every cyclic decimal rotation is also prime, grouped by orbit.
POST	/api/primes/after	List consecutive proven primes after an integer.
POST	/api/primes/batch-check	Check a list of integers. Test up to 1,000 decimal integers together.
POST	/api/primes/before	List consecutive proven primes before an integer.
POST	/api/primes/check	Check a number. Choose a quick probable-prime test or a rigorous proof.
POST	/api/primes/classify	Classify a prime. Rigorously test a prime against 56 structural, digital, sequence, and constellation classes.
POST	/api/primes/contiguous-digits	Find primes inside a number. Test every distinct contiguous decimal substring without changing digit order.
POST	/api/primes/coprimes	Explore relative primality. Compute φ(m), preview the reduced residue system, and find consecutive integers coprime to m after any starting point.
POST	/api/primes/count	Calculate π(x). Count every positive prime less than or equal to x exactly with primecount's parallel Gourdon algorithm.
POST	/api/primes/digit-constrained	Search a digit alphabet. Generate only candidates composed of selected decimal digits, then rigorously prove the primes.
POST	/api/primes/distribution	Measure density and residues. Count proven primes per interval bin and residue class while measuring twin pairs and the largest internal gap.
POST	/api/primes/factor-count-distribution	Compare ω(n) and Ω(n). Factor every integer in a finite range and compare distinct prime-factor counts with counts that include multiplicity.
POST	/api/primes/gap-statistics	Analyze prime-gap statistics. Compute exact frequencies, extrema, rational mean and median, and mode over consecutive prime gaps.
POST	/api/primes/gaps	Measure consecutive prime gaps. Report each gap with its merit, the gap over the natural logarithm of its lower prime, and highlight the largest gap in the interval.
POST	/api/primes/gaussian/check	Check a + bi. Apply the exact norm and rational-axis criteria in the Gaussian integers.
POST	/api/primes/gaussian/range	Find Gaussian primes. Search the square −B ≤ a,b ≤ B and inspect the prime lattice.
POST	/api/primes/generate	Create fixed-size primes. Generate distinct primes with an exact decimal length.
POST	/api/primes/generate-special	Generate primes with structure. Create rigorously proven safe, Sophie Germain, Blum, or modular primes.
POST	/api/primes/goldbach	Goldbach partitions of an even integer. Find unique proven-prime pairs whose sum is the supplied even integer.
POST	/api/primes/indicator-constant	Encode primality as binary digits. Calculate C = Σ [n is prime]·2⁻ⁿ with a certified tail bound and arbitrary-precision decimal output.
POST	/api/primes/integer-profile	Analyze arithmetic functions. Factor one arbitrary-size integer and calculate its divisor, totient, Carmichael, Möbius, radical, and semiprime data.
POST	/api/primes/miller-rabin-witnesses	Miller–Rabin witness bases. Use the complete strong test—not the incomplete aᵈ = 1 shortcut—to distinguish witnesses and passing bases.
POST	/api/primes/modular	Calculate modulo a prime. Calculate inverses, powers, multiplicative orders, all square roots, or a primitive root.
POST	/api/primes/modular-wheel	Modular wheel of residue structure. Plot native primality and coprimality results around modular rings.
POST	/api/primes/nth	Find the n-th prime. Look up p(n) exactly using parallel primecount, with a PARI/GP fallback.
POST	/api/primes/nth-near	Primes near a number. Find the nth prime or list consecutive primes strictly before or after an arbitrary-size integer.
POST	/api/primes/palindrome-derived	Test |n − reverse(n)| + 1. Reproduce the source sequence with exact decimal reversal and rigorous primality tests.
POST	/api/primes/paterson	Paterson primes by base-4 form. Find prime p when its base-4 digits, read as a decimal integer, are also prime.
POST	/api/primes/perfect	Generate even perfect numbers. Use the Euclid–Euler theorem with rigorously proven Mersenne primes.
POST	/api/primes/polynomial	Euler's prime polynomial n² − n + k. Find rigorous prime values, the longest consecutive run, and residue classes forced divisible by small primes.
POST	/api/primes/brun	Sum reciprocals over twin, cousin, sexy, triplet or quadruplet primes up to a bound in MPFR, reporting the exact truncated sum, the tuple count and the published estimate for comparison.
POST	/api/primes/mertens	Compute M(x) by the hyperbola identity, optionally cross-checked against an independent segmented Mobius sieve, or track every partial sum for sign changes, extrema and the largest |M(n)|/sqrt(n).
POST	/api/primes/fermat-quotients	Search a range for Wieferich (any base), Wall-Sun-Sun, Wilson or Wolstenholme primes in the compiled scanner, reporting hits, near-misses by Fermat quotient, and the range actually exhausted.
POST	/api/primes/primorials	Primorials p#. Multiply the first n rigorously generated primes cumulatively with arbitrary precision.
POST	/api/primes/problems	Run exact equation searches. Explore four problems from the source notebook with native factorizations and rigorous primality checks.
POST	/api/primes/progression	Primes in a residue class. Find proven primes p ≡ r (mod m) in an inclusive interval.
POST	/api/primes/pyramid	Construct number pyramids. Recreate the digit-insertion sequence or a multiplication pyramid with native primality flags.
POST	/api/primes/random-range	Sample proven primes. Select distinct random primes inside an arbitrary-precision interval.
POST	/api/primes/range	Primes in an interval. Results are paginated, even when the endpoints are huge.
POST	/api/primes/reciprocal	Analyze 1/p. Find the exact decimal period, test whether 10 is a primitive root modulo p, and export decimal digits.
POST	/api/primes/reptend	Full-reptend primes and maximal decimal periods. Find primes p for which 1/p has the maximum possible period p−1.
POST	/api/primes/sieve-interval	Enumerate primes in an interval of any magnitude.
POST	/api/primes/special-numbers	Explore related sequences. Search finite ranges using corrected exact definitions for pseudoprimes and prime-bearing sequences.
POST	/api/primes/tuples	Prime tuples: twins and constellations. Search by offsets from a prime base.
POST	/api/primes/verify-certificate	Verify a primality certificate. Validate a machine-readable PARI ECPP certificate independently of the result that created it.

Job queue

Method	Path	Purpose
GET	/api/queue	Show queued and running jobs in dispatch order.

Saved reports

Method	Path	Purpose
GET	/api/reports	Search the index of saved report files.

Session

Method	Path	Purpose
GET	/api/searches	List the saved job searches, most recently updated first.
POST	/api/searches	Save a named job search, replacing one that already has the name.
DELETE	/api/searches/{search_id}	Delete one saved job search; jobs and reports are untouched.
GET	/api/session	Issue the per-process request token; the only unauthenticated route.

Engine setup

Method	Path	Purpose
GET	/api/setup	Report which native engines are present, missing or being built.
POST	/api/setup/install	Build the pinned native engines after explicit browser confirmation.
GET	/api/setup/log	Return the native-engine build log.

Integer structure

Method	Path	Purpose
POST	/api/structure/factorint-strategies	Compare PARI factoring strategies. The second argument of factorint is a bitmask of methods to avoid: 1 avoids MPQS, 2 avoids the first-stage ECM, 4 avoids Pollard–Brent rho and Shanks SQUFOF, 8 skips the final ECM.
POST	/api/structure/lenstra-divisors	Search divisors with Lenstra's algorithm. PARI's divisorslenstra finds every divisor d of N with d ≡ r (mod s) without factoring N.
POST	/api/structure/predicates	Test the PARI structure predicates. Run isprimepower (which returns the exponent k of n = pᵏ), ispseudoprimepower, ispowerful, istotient (which returns a witness m with φ(m) = n), isfundamental and ispolygonal on one integer.

Independent verification

Method	Path	Purpose
POST	/api/verify/mersenne	Decide 2^p - 1 with PARI/GP's and YAFU's independent Lucas-Lehmer implementations and compare them; the test is deterministic for a prime exponent, so each verdict is a proof.
POST	/api/verify/nth-prime	Compute the n-th prime exactly with primecount and primesieve, two separate codebases, and compare them.
POST	/api/verify/primality	Decide primality with independent implementations and compare them.
POST	/api/verify/prime-count	Compare every available pi(x) method across as many as three engine implementations.
POST	/api/verify/self-test	Ask each installed engine questions with published answers.

Visualization

Method	Path	Purpose
POST	/api/visual/complexity	Compare asymptotics with measured runs.
POST	/api/visual/eisenstein-lattice	Map Eisenstein primes on the hexagonal lattice. Enumerate every Eisenstein prime a + bω with norm a² − ab + b² up to a bound and separate the split, inert, and ramified cases.
POST	/api/visual/gap-timeline	Prime gaps and maximal record gaps. Plot every consecutive prime gap across a range and mark each record gap with its merit g / ln p.
POST	/api/visual/modular-wheel	Spin a wheel with an arbitrary base. Place consecutive integers on a wheel of any base up to 10,000 from any starting point, with per-spoke prime counts and Euler's totient computed natively.
POST	/api/visual/prime-race	Prime race animation of the residue classes. Animate the running prime counts of every class coprime to m at native checkpoints and replay each exact lead change.
POST	/api/visual/residue-heatmap	Count primes by residue class and interval. Split a range into equal bins and count the primes in every residue class modulo m, exposing the reduced classes predicted by Dirichlet's theorem.
POST	/api/visual/sieve-trace	Replay a sieve step by step. Play back the exact execution trace of the sieve of Eratosthenes, its segmented variant, Sundaram's sieve, or Atkin's sieve.
POST	/api/visual/spiral	Draw a prime spiral. Lay a range of integers on an Ulam square spiral, a Sacks parabolic spiral, or an Archimedean polar spiral, optionally highlighting a residue class or a quadratic family.

Workspaces

Method	Path	Purpose
GET	/api/workspaces	List saved workspaces, newest first.
POST	/api/workspaces	Save a named workspace referencing jobs, reports, and interface state.
DELETE	/api/workspaces/{workspace_id}	Delete one workspace. Jobs and reports are not affected.
GET	/api/workspaces/{workspace_id}	Load one workspace.
POST	/api/workspaces/{workspace_id}	Update selected fields of one workspace.

Zeta and L-functions

Method	Path	Purpose
POST	/api/zeta/backlund-s	Certified S(T) with an exploratory plot across an ordinate range.
POST	/api/zeta/characters	Dirichlet character table modulo q with conductor, parity and order.
POST	/api/zeta/chebyshev-psi	Chebyshev ψ(x) reconstructed from certified zeros (FLINT/Arb).
POST	/api/zeta/count	Count nontrivial zeros. Certify the exact number of zeros from ordinate 0 through T.
POST	/api/zeta/dedekind	Dedekind zeta of a bounded number field through PARI/GP.
POST	/api/zeta/euler-product	Truncated Euler products against ζ(s) with a rigorous tail bound.
POST	/api/zeta/evaluate	Evaluate ζ(s). Compute certified complex enclosures with FLINT/Arb ball arithmetic.
POST	/api/zeta/explicit-prime-count	Riemann's explicit formula for π(x) from certified zeros (FLINT/Arb).
POST	/api/zeta/functional-equation	Verify ζ(s) symmetry. Independently enclose both sides of the functional equation and test rigorous ball overlap.
POST	/api/zeta/gram	Calculate a Gram point. Enclose gₙ defined by the Riemann–Siegel theta equation θ(gₙ)=πn.
POST	/api/zeta/gram-blocks	Gram's-law verdicts, exceptions and Gram blocks over an index range.
POST	/api/zeta/hardy	Evaluate Hardy Z(t). Evaluate the real-valued critical-line function whose real zeros correspond to ζ zeros.
POST	/api/zeta/heatmap	Render a ζ(s) heatmap. Color magnitude and phase across a sampled complex region.
POST	/api/zeta/l-function	Rigorous L(s, χ) enclosure with an independent Hurwitz cross-check.
POST	/api/zeta/l-zeros	GRH experiments on critical-line zeros of L(s, χ): certified sign changes, or exploratory |L| minima.
POST	/api/zeta/line	Plot ζ(½ + it). FLINT evaluates every point; select a component or trace the complex values in an Argand plot.
POST	/api/zeta/pair-correlation	Pair correlation of certified zeros against the GUE prediction.
POST	/api/zeta/riemann-siegel	Riemann–Siegel main sum with K corrections against acb_dirichlet_zeta.
POST	/api/zeta/stieltjes	Calculate a Stieltjes constant. Enclose γₙ from the Laurent expansion of ζ(s) around its pole at s=1.
POST	/api/zeta/xi-eta	Evaluate ξ(s) or η(s). Evaluate the completed xi function or Dirichlet eta continuation with rigorous complex balls.
POST	/api/zeta/zero-spacing	Normalized nearest-neighbour spacing histogram of certified zeros.
POST	/api/zeta/zeros	Isolate consecutive critical-line zeros. Return rigorous intervals for consecutive Hardy Z zeros, starting at any positive index.