mathxlab.experiments.prime_suite

Prime Numbers experiment suite (E014–E046).

Quickstart

from mathxlab.experiments.prime_suite import run_e014, run_e015, CommonParams

Public API

Name	Kind	Summary
CommonParams	class	Shared parameters.
ParamsE014	class	Parameters for E014.
run_e014	function	E014: Primorial ± 1: “Euclid numbers are prime” fails quickly.
ParamsE015	class	Parameters for E015.
run_e015	function	E015: Wilson’s theorem is true, but the naive ‘test’ is unusable at scale.
ParamsE016	class	Parameters for E016.
run_e016	function	E016: Trial division collapses with size; MR stays fast (counterexample to ‘simple is fine’).
ParamsE017	class	Parameters for E017.
run_e017	function	E017: Full sieve memory vs segmented sieve (counterexample to ‘O(N) is fine’).
ParamsE018	class	Parameters for E018.
run_e018	function	E018: Strong pseudoprimes: too few MR bases yields false positives (counterexample).
ParamsE019	class	Parameters for E019.
run_e019	function	E019: Prime density: pi(x) vs x/log x and error curve.
ParamsE020	class	Parameters for E020.
run_e020	function	E020: li(x) is often a better approximation than x/log x (visual counterexample).
ParamsE021	class	Parameters for E021.
run_e021	function	E021: Explicit inequality sanity checks (conditions matter).
ParamsE022	class	Parameters for E022.
run_e022	function	E022: Prime race: pi(x;4,1) vs pi(x;4,3).
ParamsE023	class	Parameters for E023.
run_e023	function	E023: Distribution of primes across residue classes mod q (finite-range bias).
ParamsE024	class	Parameters for E024.
run_e024	function	E024: Ulam spiral: primes in a spiral show diagonal structure.
ParamsE025	class	Parameters for E025.
run_e025	function	E025: Prime gaps are not monotone (counterexample to naive ‘gaps always grow’).
ParamsE026	class	Parameters for E026.
run_e026	function	E026: Gap normalization: g/log p helps compare scales.
ParamsE027	class	Parameters for E027.
run_e027	function	E027: Record prime gaps vs log^2 heuristic (Cramér-style scaling).
ParamsE028	class	Parameters for E028.
run_e028	function	E028: Jumping champions: most frequent gap in sliding windows.
ParamsE029	class	Parameters for E029.
run_e029	function	E029: Twin prime counts vs simple Hardy–Littlewood-style heuristic.
ParamsE030	class	Parameters for E030.
run_e030	function	E030: Cousin and sexy primes: compare counts of prime pairs.
ParamsE031	class	Parameters for E031.
run_e031	function	E031: Prime k-tuple admissibility: mod obstructions are real counterexamples.
ParamsE032	class	Parameters for E032.
run_e032	function	E032: Count small prime constellations (triplets/quadruplets) up to N.
ParamsE033	class	Parameters for E033.
run_e033	function	E033: Small gaps exist often, but that doesn’t make them twins (data-only counterexample).
ParamsE034	class	Parameters for E034.
run_e034	function	E034: Twin prime counts vary strongly across windows (variance counterexample).
ParamsE035	class	Parameters for E035.
run_e035	function	E035: Primes in arithmetic progressions: empirical counts in reduced residues.
ParamsE036	class	Parameters for E036.
run_e036	function	E036: Small arithmetic progressions of primes (3-term and 4-term) in ranges.
ParamsE037	class	Parameters for E037.
run_e037	function	E037: Prime-free intervals exist: n!+2,…,n!+n are all composite.
ParamsE038	class	Parameters for E038.
run_e038	function	E038: Bertrand’s postulate: for n>1 there is a prime in (n, 2n).
ParamsE039	class	Parameters for E039.
run_e039	function	E039: Sophie Germain primes and safe primes (counts in range).
ParamsE040	class	Parameters for E040.
run_e040	function	E040: Palindromic primes: even-length palindromes are divisible by 11 (counterexample).
ParamsE041	class	Parameters for E041.
run_e041	function	E041: Fermat numbers: not all are prime (counterexample at F5).
ParamsE042	class	Parameters for E042.
run_e042	function	E042: Repunit primes: many k are forced composite (counterexamples).
ParamsE043	class	Parameters for E043.
run_e043	function	E043: Pollard rho runtime varies wildly (counterexample to ‘same size => same effort’).
ParamsE044	class	Parameters for E044.
run_e044	function	E044: Solovay–Strassen vs Miller–Rabin (liar rates on random composites).
ParamsE045	class	Parameters for E045.
run_e045	function	E045: Deterministic 64-bit MR: 12 bases vs small subsets (counterexample).
ParamsE046	class	Parameters for E046.
run_e124	class	Parameters for E124.
run_e124	function	E124: Klauber triangle: prime patterns on a triangular array.
ParamsE125	class	Parameters for E125.
run_e125	function	E125: Sacks spiral: primes on an Archimedean spiral (one square per turn).
ParamsE126	class	Parameters for E126.
run_e126	function	E126: Hexagonal number spiral: primes on a hexagonal lattice spiral.

Reference

Classes

class mathxlab.experiments.prime_suite.CommonParams(seed, n_max)

Bases: object

Shared parameters.

Parameters:

• seed – Random seed.

• n_max – Upper bound for sieve-based experiments (inclusive).

class mathxlab.experiments.prime_suite.ParamsE014(k_max)

Bases: object

Parameters for E014.

Parameters:

k_max – Number of primorial steps.

class mathxlab.experiments.prime_suite.ParamsE015(n_values)

Bases: object

Parameters for E015.

Parameters:

n_values – Values of n to benchmark for Wilson-style factorial mod.

class mathxlab.experiments.prime_suite.ParamsE016(bit_sizes, samples_per_size)

Bases: object

Parameters for E016.

Parameters:

• bit_sizes – Bit sizes to benchmark.

• samples_per_size – Samples per bit size.

class mathxlab.experiments.prime_suite.ParamsE017(n_values, segment_size)

Bases: object

Parameters for E017.

Parameters:

• n_values – N values to estimate sieve memory for.

• segment_size – Segment size for segmented sieve.

class mathxlab.experiments.prime_suite.ParamsE018(n_max)

Bases: object

Parameters for E018.

Parameters:

n_max – Search range for pseudoprimes.

class mathxlab.experiments.prime_suite.ParamsE019(n_max)

Bases: object

Parameters for E019.

Parameters:

n_max – Max x for pi(x).

class mathxlab.experiments.prime_suite.ParamsE020(n_max, step)

Bases: object

Parameters for E020.

Parameters:

• n_max – Max x.

• step – Step size for numerical integration.

class mathxlab.experiments.prime_suite.ParamsE021(n_max)

Bases: object

Parameters for E021.

Parameters:

n_max – Upper bound.

class mathxlab.experiments.prime_suite.ParamsE022(n_max)

Bases: object

Parameters for E022.

Parameters:

n_max – Upper bound.

class mathxlab.experiments.prime_suite.ParamsE023(n_max, q)

Bases: object

Parameters for E023.

Parameters:

• n_max – Upper bound.

• q – Modulus.

class mathxlab.experiments.prime_suite.ParamsE024(size)

Bases: object

Parameters for E024.

Parameters:

size – Odd grid size (size x size).

class mathxlab.experiments.prime_suite.ParamsE025(n_max)

Bases: object

Parameters for E025.

Parameters:

n_max – Upper bound.

class mathxlab.experiments.prime_suite.ParamsE026(n_max, bins)

Bases: object

Parameters for E026.

Parameters:

• n_max – Upper bound.

• bins – Histogram bins.

class mathxlab.experiments.prime_suite.ParamsE027(n_max)

Bases: object

Parameters for E027.

Parameters:

n_max – Upper bound.

class mathxlab.experiments.prime_suite.ParamsE028(n_max, window, step)

Bases: object

Parameters for E028.

Parameters:

• n_max – Upper bound.

• window – Number of consecutive gaps per window.

• step – Window step.

class mathxlab.experiments.prime_suite.ParamsE029(n_max)

Bases: object

Parameters for E029.

Parameters:

n_max – Upper bound.

class mathxlab.experiments.prime_suite.ParamsE030(n_max, d_values)

Bases: object

Parameters for E030.

Parameters:

• n_max – Upper bound.

• d_values – Differences to count (e.g., 4 and 6).

class mathxlab.experiments.prime_suite.ParamsE031(max_mod)

Bases: object

Parameters for E031.

Parameters:

max_mod – Check admissibility against primes up to this modulus.

class mathxlab.experiments.prime_suite.ParamsE032(n_max)

Bases: object

Parameters for E032.

Parameters:

n_max – Upper bound.

class mathxlab.experiments.prime_suite.ParamsE033(n_max, threshold)

Bases: object

Parameters for E033.

Parameters:

• n_max – Upper bound.

• threshold – Gap threshold.

class mathxlab.experiments.prime_suite.ParamsE034(n_max, window, step)

Bases: object

Parameters for E034.

Parameters:

• n_max – Upper bound.

• window – Window length.

• step – Step.

class mathxlab.experiments.prime_suite.ParamsE035(n_max, q)

Bases: object

Parameters for E035.

Parameters:

• n_max – Upper bound.

• q – Modulus.

class mathxlab.experiments.prime_suite.ParamsE036(n_max, max_d)

Bases: object

Parameters for E036.

Parameters:

• n_max – Upper bound.

• max_d – Max step d to search.

class mathxlab.experiments.prime_suite.ParamsE037(n_values)

Bases: object

Parameters for E037.

Parameters:

n_values – Values of n to demonstrate prime-free runs of length n-1.

class mathxlab.experiments.prime_suite.ParamsE038(n_max)

Bases: object

Parameters for E038.

Parameters:

n_max – Check Bertrand postulate for n up to n_max.

class mathxlab.experiments.prime_suite.ParamsE039(n_max)

Bases: object

Parameters for E039.

Parameters:

n_max – Upper bound for p.

class mathxlab.experiments.prime_suite.ParamsE040(digits_max, limit)

Bases: object

Parameters for E040.

Parameters:

• digits_max – Max digits to scan palindromes.

• limit – Stop scanning after this many palindromes.

class mathxlab.experiments.prime_suite.ParamsE041(m_max)

Bases: object

Parameters for E041.

Parameters:

m_max – Max Fermat index m to test.

class mathxlab.experiments.prime_suite.ParamsE042(k_max)

Bases: object

Parameters for E042.

Parameters:

k_max – Max repunit length k.

class mathxlab.experiments.prime_suite.ParamsE043(samples, bits)

Bases: object

Parameters for E043.

Parameters:

• samples – Number of semiprimes to factor.

• bits – Bit-size for factors.

class mathxlab.experiments.prime_suite.ParamsE044(samples, bits, bases)

Bases: object

Parameters for E044.

Parameters:

• samples – Random odd integers to test.

• bits – Bit size.

• bases – Bases per test.

class mathxlab.experiments.prime_suite.ParamsE045(samples, bits)

Bases: object

Parameters for E045.

Parameters:

• samples – Random odd integers to test.

• bits – Bit size.

class mathxlab.experiments.prime_suite.ParamsE046(n_max, mr_bases)

Bases: object

Parameters for E046.

Parameters:

• n_max – Range end.

• mr_bases – Base set for fast MR stage.

class mathxlab.experiments.prime_suite.ParamsE124(size)

Bases: object

Parameters for E124.

Parameters:

size – Grid size (odd). The triangle visualizes integers 1..size^2.

class mathxlab.experiments.prime_suite.ParamsE125(size)

Bases: object

Parameters for E125.

Parameters:

size – Scale parameter (odd). The experiment visualizes integers 1..size^2.

class mathxlab.experiments.prime_suite.ParamsE126(size)

Bases: object

Parameters for E126.

Parameters:

size – Scale parameter (odd). The experiment visualizes integers 1..size^2.

Functions

mathxlab.experiments.prime_suite.run_e014(*, out_dir, seed, figures_dir, report_path, params_path)

E014: Primorial ± 1: “Euclid numbers are prime” fails quickly.

mathxlab.experiments.prime_suite.run_e015(*, out_dir, seed, figures_dir, report_path, params_path)

E015: Wilson’s theorem is true, but the naive ‘test’ is unusable at scale.

mathxlab.experiments.prime_suite.run_e016(*, out_dir, seed, figures_dir, report_path, params_path)

E016: Trial division collapses with size; MR stays fast (counterexample to ‘simple is fine’).

mathxlab.experiments.prime_suite.run_e017(*, out_dir, seed, figures_dir, report_path, params_path)

E017: Full sieve memory vs segmented sieve (counterexample to ‘O(N) is fine’).

mathxlab.experiments.prime_suite.run_e018(*, out_dir, seed, figures_dir, report_path, params_path)

E018: Strong pseudoprimes: too few MR bases yields false positives (counterexample).

mathxlab.experiments.prime_suite.run_e019(*, out_dir, seed, figures_dir, report_path, params_path, n_max=2000000)

E019: Prime density: pi(x) vs x/log x and error curve.

mathxlab.experiments.prime_suite.run_e020(*, out_dir, seed, figures_dir, report_path, params_path, n_max=3000000)

E020: li(x) is often a better approximation than x/log x (visual counterexample).

mathxlab.experiments.prime_suite.run_e021(*, out_dir, seed, figures_dir, report_path, params_path, n_max=2000000)

E021: Explicit inequality sanity checks (conditions matter).

mathxlab.experiments.prime_suite.run_e022(*, out_dir, seed, figures_dir, report_path, params_path)

E022: Prime race: pi(x;4,1) vs pi(x;4,3).

mathxlab.experiments.prime_suite.run_e023(*, out_dir, seed, figures_dir, report_path, params_path)

E023: Distribution of primes across residue classes mod q (finite-range bias).

mathxlab.experiments.prime_suite.run_e024(*, out_dir, seed, figures_dir, report_path, params_path, size=301)

E024: Ulam spiral: primes in a spiral show diagonal structure.

mathxlab.experiments.prime_suite.run_e025(*, out_dir, seed, figures_dir, report_path, params_path)

E025: Prime gaps are not monotone (counterexample to naive ‘gaps always grow’).

mathxlab.experiments.prime_suite.run_e026(*, out_dir, seed, figures_dir, report_path, params_path)

E026: Gap normalization: g/log p helps compare scales.

mathxlab.experiments.prime_suite.run_e027(*, out_dir, seed, figures_dir, report_path, params_path)

E027: Record prime gaps vs log^2 heuristic (Cramér-style scaling).

mathxlab.experiments.prime_suite.run_e028(*, out_dir, seed, figures_dir, report_path, params_path)

E028: Jumping champions: most frequent gap in sliding windows.

mathxlab.experiments.prime_suite.run_e029(*, out_dir, seed, figures_dir, report_path, params_path)

E029: Twin prime counts vs simple Hardy–Littlewood-style heuristic.

mathxlab.experiments.prime_suite.run_e030(*, out_dir, seed, figures_dir, report_path, params_path)

E030: Cousin and sexy primes: compare counts of prime pairs.

mathxlab.experiments.prime_suite.run_e031(*, out_dir, seed, figures_dir, report_path, params_path)

E031: Prime k-tuple admissibility: mod obstructions are real counterexamples.

mathxlab.experiments.prime_suite.run_e032(*, out_dir, seed, figures_dir, report_path, params_path)

E032: Count small prime constellations (triplets/quadruplets) up to N.

mathxlab.experiments.prime_suite.run_e033(*, out_dir, seed, figures_dir, report_path, params_path)

E033: Small gaps exist often, but that doesn’t make them twins (data-only counterexample).

mathxlab.experiments.prime_suite.run_e034(*, out_dir, seed, figures_dir, report_path, params_path)

E034: Twin prime counts vary strongly across windows (variance counterexample).

mathxlab.experiments.prime_suite.run_e035(*, out_dir, seed, figures_dir, report_path, params_path)

E035: Primes in arithmetic progressions: empirical counts in reduced residues.

mathxlab.experiments.prime_suite.run_e036(*, out_dir, seed, figures_dir, report_path, params_path)

E036: Small arithmetic progressions of primes (3-term and 4-term) in ranges.

mathxlab.experiments.prime_suite.run_e037(*, out_dir, seed, figures_dir, report_path, params_path)

E037: Prime-free intervals exist: n!+2,…,n!+n are all composite.

mathxlab.experiments.prime_suite.run_e038(*, out_dir, seed, figures_dir, report_path, params_path)

E038: Bertrand’s postulate: for n>1 there is a prime in (n, 2n).

mathxlab.experiments.prime_suite.run_e039(*, out_dir, seed, figures_dir, report_path, params_path)

E039: Sophie Germain primes and safe primes (counts in range).

mathxlab.experiments.prime_suite.run_e040(*, out_dir, seed, figures_dir, report_path, params_path)

E040: Palindromic primes: even-length palindromes are divisible by 11 (counterexample).

mathxlab.experiments.prime_suite.run_e041(*, out_dir, seed, figures_dir, report_path, params_path)

E041: Fermat numbers: not all are prime (counterexample at F5).

mathxlab.experiments.prime_suite.run_e042(*, out_dir, seed, figures_dir, report_path, params_path)

E042: Repunit primes: many k are forced composite (counterexamples).

mathxlab.experiments.prime_suite.run_e043(*, out_dir, seed, figures_dir, report_path, params_path)

E043: Pollard rho runtime varies wildly (counterexample to ‘same size => same effort’).

mathxlab.experiments.prime_suite.run_e044(*, out_dir, seed, figures_dir, report_path, params_path)

E044: Solovay–Strassen vs Miller–Rabin (liar rates on random composites).

mathxlab.experiments.prime_suite.run_e045(*, out_dir, seed, figures_dir, report_path, params_path)

E045: Deterministic 64-bit MR: 12 bases vs small subsets (counterexample).

mathxlab.experiments.prime_suite.run_e046(*, out_dir, seed, figures_dir, report_path, params_path)

E046: A practical prime-testing pipeline (and how it can fail if tuned badly).

mathxlab.experiments.prime_suite.run_e124(*, out_dir, seed, figures_dir, report_path, params_path, size=301)

E124: Klauber triangle: prime patterns on a triangular array.

Notes

The Klauber triangle (1932) places row n as the consecutive integers (n-1)^2+1 .. n^2 (so each row has length 2n-1). When primes are highlighted, lines rich in primes become visible (often associated with quadratic polynomials), similar in spirit to the Ulam spiral.

This experiment is deterministic; seed is accepted for framework consistency but does not affect the output.

mathxlab.experiments.prime_suite.run_e125(*, out_dir, seed, figures_dir, report_path, params_path, size=301)

E125: Sacks spiral: primes on an Archimedean spiral (one square per turn).

Notes

In the Sacks spiral, integer n is placed at polar coordinates:

r = sqrt(n), θ = 2π r

so perfect squares lie on the positive x-axis (θ is a multiple of 2π). Primes are highlighted to reveal curve-like prime concentrations.

This experiment is deterministic; seed is accepted for framework consistency but does not affect the output.

mathxlab.experiments.prime_suite.run_e126(*, out_dir, seed, figures_dir, report_path, params_path, size=301)

E126: Hexagonal number spiral: primes on a hexagonal lattice spiral.

Notes

This is a hex-grid analogue of the Ulam spiral: integers are placed in a spiral on a hexagonal tiling and primes are highlighted.

The parameter size controls the max integer via n_max = size^2 so the prime counts are comparable to square-grid experiments.

This experiment is deterministic; seed is accepted for framework consistency but does not affect the output.