Modular Arithmetic vs Millennium Problems

A calculator for arithmetic modulo N. It lets you choose a fixed modulus, and then make lots of calculations without having to press a "mod" button again and again. It also: - follows the order convention; - supports arbitrarily large numbers; - performs fast modular division and exponentiation; - can show a full transcript of your calculation. Modular arithmetic is a "calculus of remainders". It features throughout mathematics and computer science, and has applications from cryptography to barcodes to music. The basic idea is that you choose a modulus N, and then reduce every number to one of the integers 0,1,2,...,N−1 according to what remainder it leaves when dividing by N. For example, using a modulus of 17: 40 ≡ 6 (since 40 ÷ 17 leaves a remainder of 6); 17 ≡ 0 (since 17 ÷ 17 leaves no remainder at all). Arithmetic follows these same rules. Still using a modulus of 17: 15 + 7 ≡ 5 (since 22 ≡ 5); 3 × 9 ≡ 10 (since 27 ≡ 10); 5 ^ 3 ≡ 6 (since 125 ≡ 6). Subtraction and division behave in a way that complements addition and multiplication: −1 ≡ 16 (since 16 + 1 = 17 ≡ 0); 1/2 ≡ 9 (since 9 × 2 = 18 ≡ 1); 4 - 7 ≡ 14 (since 14 + 7 = 21 ≡ 4); 7 ÷ 3 = 8 (since 8 × 3 = 24 ≡ 7). There are no negative numbers or fractions: like −1 and 7 ÷ 3 in the examples above, these are also reduced to one of 0,1,...,N−1. As usual, you cannot divide by zero. You also cannot divide if the right hand side has any common factors with the modulus. If we change our modulus to 10, then the following operations all generate errors: 3 ÷ 20 (since 20 ≡ 0); 7 ÷ 8 (since 8 and 10 have a common factor of 2). Integers can be arbitrarily large. For instance, if we set our modulus to 2305843009213693951 (a Mersenne prime), then: 5 ^ 2305843009213693950 ≡ 1 (by Fermat's little theorem). The code is written carefully, and is backed up by a thorough suite of 186 automated tests. This app supports external keyboards, Siri Shortcuts, and (on iPad) Slide Over, Split View, and multiple windows.

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