Millennium Problems vs p

Can you solve a problem worth $1,000,000? The Clay Mathematics Institute has offered a million-dollar prize for each of seven unsolved mathematical problems — and nobody has claimed them yet. Millennium Problems brings these mysteries to life for everyone, from curious 11-year-olds to doctoral researchers. SEVEN LEGENDARY PROBLEMS • P vs NP — Is finding as hard as checking? • Riemann Hypothesis — Where do prime numbers hide? • Yang-Mills & Mass Gap — Why do particles have mass? • Navier-Stokes — Can we predict turbulent fluids? • Hodge Conjecture — What shapes are built from algebra? • Birch & Swinnerton-Dyer — How many solutions does an equation have? • Poincaré Conjecture — The shape of the universe (SOLVED!) EVERY SKILL LEVEL Choose your grade — from Grade 6 to Graduate — and the content adapts instantly. The same problem is explained with simple analogies for a middle schooler and rigorous proofs for a PhD student. INTERACTIVE 3D VISUALISATIONS Every problem comes with a real-time Three.js visualisation. Watch elliptic curves rotate, Riemann zeros appear on the critical line, and fluid particle fields swirl — all rendered natively on your device. BUILT-IN QUIZZES Test your understanding with 5 questions per problem per grade level. Earn XP, track your streak, and collect badges as you explore. WORKS OFFLINE No account. No subscription. No ads. Everything runs locally on your device. OPTIONAL BIOMETRIC LOCK Protect your progress with Face ID, Touch ID, or fingerprint — entirely optional. Whether you're a student looking for a challenge, a teacher seeking engaging material, or a mathematician revisiting the classics, Millennium Problems meets you where you are. Clay Mathematics Institute · $7,000,000 in unclaimed prizes · Since 2000

p-adelic shows a number in all of its completions at once. Type 1/3, √2, a polynomial or log 7 and watch the adelic strip: one column for the real line and one for each prime p, each streaming the number's digits in base p and growing leftward as you turn the precision dial. There is no "p-adic mode" to switch into — you always see the whole picture, the way number theorists think about it. THE STRIP • A column per place: ∞ plus the primes that matter for your input, and any prime you pin. • Streaming precision from 5 to 200 digits; every column updates together. • Exact periodicity for rationals: the repeating block is marked with an overline, and long runs of digits fold into pills you can tap open. • Tap a column to focus it; the focused card shows valuations, unit parts, Legendre symbols and the Laurent form. DEEP DIVE • Digits, Laurent series and Witt-vector views of the same number. • Nested balls: zoom into ℤₚ ball by ball, following your number's digits. • The Bruhat–Tits tree and the Berkovich line of the chosen place. ALGEBRAIC NUMBERS • √d, roots of unity ζₙ, Teichmüller representatives, and any algebraic number given by its minimal polynomial, e.g. algebraic(x³ − 2). • See whether a prime splits, stays inert or ramifies; extend the field with one tap and cycle through the Galois conjugates. POLYNOMIALS • Newton polygons at every prime and at ∞, with slopes, multiplicities and the Krasner check. SPECIAL FUNCTIONS • p-adic exp, log and Artin–Hasse, with their convergence disks. • Factorials with Legendre's formula, Morita's Γₚ with the Wilson check. • Kubota–Leopoldt ζₚ and L-values, Hilbert symbols with the reciprocity check, and Tate's local factors. LEARN AND REUSE • A Library of worked examples for every idea above, from −1 = …(p−1)(p−1)(p−1) to the Berkovich line of ∛2. • History you can reopen with one tap. • VoiceOver describes every visualization; text and digits follow Dynamic Type. PRIVATE BY DESIGN Everything is computed on your device. The app has no account, no analytics and makes no network connections. For students meeting p-adic numbers for the first time, teachers who want a live picture to point at, and researchers who need a Newton polygon or a Hilbert symbol checked in seconds.