Mersenne Listed 2^67 − 1 as Prime. In 1903, Cole Found Its Factors.

In 1644 Mersenne put 2^67 − 1 on his list of numbers he believed were prime. In 1903 Frank Nelson Cole found its factors. This video follows the numbers and the documents: Mersenne's list, what the Lucas–Lehmer test can and cannot tell you, Lucas's 1891 remark (right about 2^67 − 1, wrong about 2^89 − 1), what Cole's paper actually prints — and how the famous silent-blackboard story compares with the record. 💬 Two to the 67th, minus one, has 21 decimal digits. How many decimal digits does 2^127 − 1 have? Answer in the comments. Short version:    • Shorts   0:00 Mersenne's list and Cole's factors 0:12 Mersenne numbers, 1644 0:42 Prime or not? The test is not enough 1:20 Lucas, 1891 1:48 New York, 31 October 1903 2:04 How Cole searched 3:32 The famous story vs the record 4:06 Today 4:30 Your turn Notes. The blackboard scenes are modern re-enactments; the printed page shown is Lucas 1891, p. 376. The silent-blackboard story, the applause and "three years of Sundays" come from E. T. Bell's later account (1951); the Bulletin's report of the meeting mentions none of them, and historians treat many of Bell's anecdotes cautiously. The RSA paper says the system's security rests in part on the difficulty of factoring. Sources: F. N. Cole, "On the factoring of large numbers", Bulletin of the American Mathematical Society 10 (1903), pp. 134–137 (read before the Society on 31 October 1903). "The October Meeting of the American Mathematical Society", Bulletin AMS 10 (1904), pp. 171–178. É. Lucas, Théorie des nombres, vol. 1 (1891), p. 376 (Internet Archive scan, public domain). R. E. Powers, "The Tenth Perfect Number", American Mathematical Monthly (1911). R. L. Rivest, A. Shamir, L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Communications of the ACM 21 (1978). E. T. Bell, Mathematics: Queen and Servant of Science (1951), p. 228, as quoted by Leo Corry, Hunting Prime Numbers — from Human to Electronic Computers, The Rutherford Journal (later account of the silent calculation). Numbers checked with verification scripts: 2^67 − 1 = 193,707,721 × 761,838,257,287 (both prime); A = 381,015,982,504, B = 380,822,274,783; Lucas–Lehmer sequences for 2^7 − 1 and 2^11 − 1; 5959 = 80² − 21² = 59 × 101. The Pollard rho timing (about 3 ms) is our own measurement on our own server. #mathematics #historyofmathematics #primenumbers #mersenne #cole #numbertheory