One failing case refutes a for-all claim; any number of passing cases leaves it unproved.
Question: What does "for all" commit you to?
Re-simulated Re-simulated: exact integer arithmetic recomputed by data/scripts/c03.s01.L1.py, which tests each value for primality and factors each first failure. The frames walk five universal claims case by case: Euler's polynomial passes 40 cases and fails at n = 40 with 1681 = 41 × 41; Fermat's numbers pass 5 and fail at k = 5 with 4294967297 = 641 × 6700417; Mersenne's claim and the odd-number claim fall at p = 11 and m = 9; n³ − n passes all 61 tested cases and is still not proved.
Frame 1 of 12: A claim about every caseEuler's polynomial says: n² + n + 41 is prime for every whole number n. No case has been tested yet, so the row of dots is empty and nothing here is evidence.
A claim about every case. Euler's polynomial says: n² + n + 41 is prime for every whole number n. No case has been tested yet, so the row of dots is empty and nothing here is evidence.
Keyboard: Space plays or pauses, Left and Right step, Home goes to the first frame, End goes to the last frame.
Text description of this lab
A row of dots counts the cases a universal claim has survived. Gold filled dots are cases that passed; the first failure is a russet ring, and the line below gives its input, value and factors. Euler's polynomial shows 40 passes, then 1681 = 41 × 41. Fermat's numbers show 5 passes, then 4294967297 = 641 × 6700417. The claim about n³ − n shows 61 passes and still reads not proved.