Valid Tags

This page defines the allowed tags for experiments in py-mathx-lab. Tags are used in the Experiments Gallery and individual experiment pages to categorize content.

Primary Tags (Domains)

These represent the broad mathematical area of the experiment.

Tag	Description
analysis	Calculus, real/complex analysis, limits, and approximation.
aps	Arithmetic Progressions and related theorems.
conjecture-generation	Patterns suggest statements that might be true (or false).
counterexample-search	Systematic exploration tries to break a hypothesis early.
dirichlet-characters	Dirichlet characters modulo \(q\), orthogonality, and primitive characters.
l-functions	Dirichlet \(L\)-functions and related analytic objects controlling prime distribution.
model-checking	Validate (or invalidate) approximations and heuristics.
number-theory	Properties of integers, divisibility, and prime numbers.
prime-races	Prime number races (e.g., Chebyshev bias) comparing counts in residue classes.
quantitative-exploration	Estimate constants, rates, limits, or distributions.
visualization	Reveal structure that is hard to see symbolically.

Secondary Tags (Topics & Methods)

These provide more specific detail about the techniques or subtopics involved.

Tag	Description
arithmetic-functions	Classical functions on integers (e.g., \(\phi\), \(\mu\), \(\sigma\), \(\tau\)) and their relations.
bounds	Explicit mathematical bounds (e.g., on prime-counting functions).
carmichael-lambda	Carmichael’s lambda function λ(n), the exponent of (Z/nZ)*.
carmichael	Carmichael numbers (absolute Fermat pseudoprimes).
classification	Grouping objects into classes based on shared properties.
critical-line	Zeta/L-function values on Re(s)=1/2 and related numerics.
dirichlet-convolution	Dirichlet convolution of arithmetic functions and identity checks.
dirichlet-series	Dirichlet generating functions and related analytic tools.
divisor-function	Divisor functions such as τ(n)=d(n) and σ(n), including record behavior.
explicit-formula	Explicit formulas connecting primes and zeros (ψ(x), π(x), etc.).
exploration	Open-ended search for patterns or properties.
factorization	Integer factorization methods and hardness.
fermat	Fermat numbers and Fermat primes.
gaps	Studies of the distribution of gaps between primes.
generating-functions	Ordinary/exponential generating functions (esp. partitions).
gram-points	Gram points and Gram’s law heuristics for zeta zeros.
hardy-z	Hardy’s Z-function and Riemann–Siegel theta function.
heuristics	Probabilistic or empirical models (e.g., Cramér’s model for primes).
li-x	Logarithmic integral \(\operatorname{li}(x)\) and its relation to \(\pi(x)\).
liouville	Liouville function λ(n) and related parity questions for Ω(n).
lucas-lehmer	Lucas–Lehmer primality test for Mersenne numbers.
mangoldt	von Mangoldt function Λ(n) and related Chebyshev functions θ(x), ψ(x).
mersenne	Mersenne numbers \(M_p = 2^p - 1\) and Mersenne primes.
miller-rabin	Miller–Rabin primality test and its counterexamples.
mobius	Möbius function \(\mu(n)\), Möbius inversion, squarefree indicators.
multiplicative	Multiplicative arithmetic functions; Dirichlet convolution viewpoints.
numerics	Heavy use of floating-point or high-precision computation.
omega	Prime-factor counting functions ω(n) and Ω(n) (distinct vs with multiplicity).
open-problems	Related to famous unproven conjectures.
optimization	Finding maxima, minima, or best-fit parameters.
partition	Partition function p(n), identities, and asymptotics.
perfect	Related specifically to perfect, abundant, or deficient numbers.
pnt	Prime Number Theorem and related asymptotics.
primes	Prime number distribution, density, and related theorems.
primorial	Primorials, Euclid numbers, primorial primes.
pseudoprime	Pseudoprimes, primality-test failures.
pseudoprimes	Collective study of various types of pseudoprimes.
riemann-zeta	Riemann zeta function ζ(s): series, Euler product, analytic continuation, zeros.
search	Systematic search through a large state space.
semiprime	Semiprimes, RSA-type composites.
sieving	Sieve methods (Eratosthenes, Sundaram, Atkin, etc.).
sigma	Related to the sum-of-divisors function \(\sigma(n)\).
summatory	Summatory functions (e.g., Mertens \(M(x)\), summatory totient \(\Phi(x)\)).
taylor	Related to Taylor series and their approximations.
totient	Euler’s totient function \(\phi(n)\), totient equations, summatory behavior.
twin	Twin primes and the twin prime conjecture.
klauber	Klauber triangle and prime patterns between consecutive squares.
sacks	Sacks spiral (\(r=\sqrt{n},\ \theta=2\pi\sqrt{n}\)) and related prime patterns.
hex-spiral	Integer spirals on a hexagonal lattice (hex-grid prime maps).
ulam	Ulam spiral and related prime patterns in 2D.
wieferich	Wieferich primes and related congruences.
wilson	Wilson’s theorem and Wilson primes.
zeta-zeros	Nontrivial zeros, zero-counting \(N(T)\), root bracketing.

Usage

When adding a new experiment:

1. Choose at least one Primary Tag (Domain or Type).

2. Choose one or more Secondary Tags (Topics & Methods).

3. Add them to the **Tags:** line in your .md file.

4. Update the Experiments Gallery using the corresponding CSS classes (tag-primary for primary tags, tag-secondary for secondary tags).