Prime Sieve (Eratosthenes)
An ancient elimination algorithm: write all integers, circle 2 and strike out all multiples of 2; circle the next survivor (3) and strike its multiples; repeat. What endures are the primes.
Module I · Linear Spectrum
Number-Line Sieve & Resonance
Explore prime distribution across the number line. Filter by modular residues (e.g. Dirichlet's arithmetic progressions $4k+1$ vs $4k+3$) or inspect gap sizes between consecutive primes.
Primes in view: 0
Max gap: 0
Twin pairs: 0
Hover any integer point to view prime factorisation, twin distance, and modular properties.
Module II · Diagonal Geometry
The Ulam Spiral & Polynomial Roads
In 1963, Stanislaw Ulam doodled integers spiraling outward on grid paper during a boring lecture. Diagonal alignments sprang forth immediately: Euler's polynomial $n^2 + n + 41$ produces 40 consecutive primes, carving gleaming constellations in the grid.
Β§
Cipher's Secret Note on Spirals
βTo the naive cryptanalyst, primes look like Gaussian white noise β a keystream generated by a fickle deity. But turn the coordinate system into an Archimedean or rectangular spiral, and rays crystallise.
These diagonal streaks reflect quadratic polynomials $an^2 + bn + c$ that happen to have an unusually dense harvest of prime values, governed by the Bateman-Horn conjecture and class numbers of imaginary quadratic fields.
Euler's Masterpiece:
$f(n) = n^2 + n + 41$
Evaluates to primes for $n = 0, 1, 2, \dots, 39$. Notice the uninterrupted luminous line piercing across our Ulam lattice!
$f(n) = n^2 + n + 41$
Evaluates to primes for $n = 0, 1, 2, \dots, 39$. Notice the uninterrupted luminous line piercing across our Ulam lattice!
Module III · The Music of Primes
The Riemann Hypothesis & Spectral Zeros
In 1859, Bernhard Riemann connected the discrete stairs of primes $\pi(x)$ to the zeros of the analytic continuation of $\zeta(s) = \sum n^{-s}$. All non-trivial zeros seem to sit on the critical line $\mathrm{Re}(s) = 1/2$.
The Explicit Formula
Riemann revealed that the prime counting staircase is an exact sum of smooth logarithmic integrals minus a wave for every zero $\rho = \tfrac{1}{2} + i\gamma_k$ of the zeta function:
$\psi(x) \approx x - \sum_{\rho} \frac{x^\rho}{\rho} - \ln(2\pi)$
Each zeta zero is an acoustic harmonic. When all zeros sing in phase on the line $1/2$, the fluctuations never exceed $\mathcal{O}(\sqrt{x} \ln x)$. If even one zero slipped off the line, primes would fluctuate with violent, asymmetric spikes.
5 zeros
First 5 Gram Zeros $\gamma_k$:
14.1347, 21.0220, 25.0108, 30.4248, 32.9350
Reconstruction of Prime Escalier $\psi(x)$
Gold stairs: exact prime counting step function. Cyan wave: acoustic reconstruction via the selected non-trivial zeta zeros.
Module IV · The Oracle's Game
Predict the Next Prime & Gap Hunter
Put your prime intuition to the test. Step into a cryptic interval and predict where the next prime lands or whether the next integer is composite. Can you beat the Prime Number Theorem estimate $\ln(n)$?
Current Anchor Prime
199
Expected Average Gap ($\approx \ln P$)
5.29
Player Intuition Score
0 pts
What is the gap to the very next prime after 199?
Make your guess above to advance the frontier.
Codex · Plain Language
The Cryptographer's Lexicon
No obscure academic walls: deciphering the language of prime theory in lucid terms.
Twin Prime Conjecture
Pairs of primes separated by only two integers, like $(11, 13)$ or $(41, 43)$. Yitang Zhang proved in 2013 that primes repeat with gaps bounded under 70 million infinitely often (now down to 246).
Riemann Hypothesis ($1859$)
States that all non-trivial zeros of the Riemann Zeta Function have a real part equal to $1/2$. If true, prime distribution has the lowest possible noise, perfectly bound by square root deviations.
Cramer's Model & Prime Gaps
A probabilistic model viewing integers $n$ as independent coins landing heads (prime) with probability $1 / \ln n$. Harald Cramér conjectured that maximal gaps scale like $\mathcal{O}((\ln n)^2)$.
Dirichlet's Theorem
Any arithmetic progression $a, a+q, a+2q, \dots$ where $\gcd(a, q) = 1$ contains infinitely many primes. For modulus 4, primes split between forms $4k+1$ (sums of two squares) and $4k+3$.
Euler's Lucky Numbers
Numbers $A$ such that $n^2 + n + A$ produces primes for all $0 \le n \le A-2$. The greatest such lucky integer is 41, linked directly to the Heegner number 163.