# mrlynum

The instruments of Mrly: primes, divisors, series, recurrences, spectra, lattices, networks. Plain numbers and byte grids go in, and counts, fractions, rates and measurements come out; every answer is exact where the integers allow and a stated approximation where they do not.

One arm reads number: primes with their ranks, gaps and square shapes, factorizations with their divisors and Mobius values, the classic sequences, and the exact rule hidden in any sequence handed over. The other arm measures shape: the tallies of a grid, the network of its filled cells, the Fourier spectrum of a field, the Walsh spectrum of a boolean function, and the visible lattice its fractions draw.

## Instruments

- **prime** tests, lists and studies primes: rank, gap, twin, and the rectangles, splits and squares a number makes.
- **factor** splits a number into primes, divisors, totient, radical and Mobius value.
- **design** enumerates the digit designs on the integer line, meters their Mobius values, splits the meter into its density echo and residual, and reads the ordinates its log-uniform spectrum carries against the zeta zeros and the design's own pole lattice.
- **classics** lists the evens, odds, powers of two, Fibonacci, Catalan and prime sequences, with exact gcd and fraction reduction.
- **blend** adds, convolves, shifts and differences sequences, then recovers the exact recurrence and its growth rate.
- **series** sums harmonic, zeta, beta and Dirichlet series, counts visible lattice points, and pins the Bernoulli fractions.
- **sieve** builds the Wallis sieve and its solid and constant-schedule cousins as schedule words, counts their cells and punctures, and prints the area, the box exponent and the limits in closed form.
- **lattice** lists the reduced fractions of a window as nodes, counts coprime pairs, and estimates pi from them.
- **census** counts the cells of a grid and its exposed faces.
- **graph** lifts a grid into a network of nodes and branches and takes its census: tips, junctions, components and fractal dimension.
- **fft** transforms in one and two dimensions and centres a field's magnitude spectrum.
- **boolean** takes a truth table's Walsh spectrum, nonlinearity, balance and avalanche.
- **gauss** classifies the Gaussian and the Eisenstein integers as units, split, inert or ramified primes, counts a symmetric window, and weighs the norm shells.
- **spirograph** takes a byte grid as a wheel with a pencil in every chosen cell, rolls it without slipping on a line, a circle or a polygon, traces every pencil, and counts the distinct curves by the coincidence law.
- **apollonian** grows an integral circle packing from its root quadruple by the Descartes reflection in exact integers, reads the circles resting on the strip's line as the Ford circles, one to every reduced fraction, and hangs the Farey stack's nodes under their tangency points.
- **spiral** winds the whole numbers on the square or the hexagonal lattice, both ways, marks the primes, twins, squarefree numbers or Mobius signs, and reads a quadratic along its diagonal.
- **zeta** walks zeta along the critical line by Euler-Maclaurin and Riemann-Siegel, finds its zeros between Gram points, and rebuilds the prime staircase from them by the explicit formula.
- **spin** spins a raster about its centre: the exact circle means and harmonics, the radial profile, the wheel it paints, and the radial stacks of turned copies.