Every Conjecture Here Is False

Every claim here is false. Your job is to break it.

Try Euler's prime generator: n² + n + 41 is prime at n = 0, 1, 10, and every value through 39. At n = 40 it becomes 1681 = 41 × 41. The boss is harder: gcd(n¹⁷ + 9, (n+1)¹⁷ + 9) stays 1 through n = 10⁵⁰. Its first counterexample has 52 digits.

What it does

This browser puzzle has 12 mathematical patterns, from warm-up to boss. Each card shows a claim, the examples that make it convincing, and a place to enter a counterexample. The app checks your input and explains either why the claim survives or why it fails. A three-step hint ladder helps you reach the harder answers, and progress is saved in your browser.

Try it in 60 seconds

1. Open the live demo. No login or installation is needed.
2. The app opens on Euler's prime generator. Enter 10 and press Break it: 151 is prime, so the claim survives.
3. Enter 40: the app stamps the claim FALSE and shows 41 × 41.
4. Press Next: the 52-digit liar, reveal all three hints, paste the number from the final hint, and press Break it.

How we built it

Each claim has its own verifier. The code uses native JavaScript BigInt for exact arithmetic, divisibility, modular exponentiation, gcd, primality checks, and cyclotomic polynomials. The 52-digit result is backed by a Python proof script: it computes the two polynomials' resultant, certifies the prime factor, and finds their unique common root. TypeScript and Vite power the interface; GitHub Actions tests and deploys it to GitHub Pages. The public source includes the tests and proof script.

Challenges and lessons

The evidence had to be accurate even when it was misleading. We tested the small cases, corrected several historical and mathematical claims during review, and limited inputs so a large number does not stall the page. The central lesson is that a long run of examples is evidence, not proof.

AI assistance

Devin by Cognition assisted with code, tests, copy, and QA. Hyperbloom explicitly allows AI-assisted development. We independently reran the 43 tests, typecheck, production build, and the boss proof script.

What's next

A mode for players to submit their own traps, and a classroom view for teachers.

Built With

• bigint
• github
• python
• sympy
• typescript
• vite
• vitest