Double a Prime, Add One, Stay Prime

Take any prime, double it, and add one. Sometimes the result is prime again, and sometimes it falls apart. Five gives eleven and survives; seven gives fifteen, which is three times five, and fails. The primes that pass this one small test are called Sophie Germain primes, and they turn out to be far more than a curiosity. Sophie Germain used exactly this condition, two hundred years ago, to prove the first real case of Fermat's Last Theorem. The same primes now sit inside the encryption that protects your traffic online. This video walks the whole story: which primes survive being doubled, why the rule that breaks most of them is as simple as landing on a multiple of three, the chains where survivors stack up (two, five, eleven, twenty-three, forty-seven), the woman who signed her work "Monsieur Le Blanc" to be taken seriously, and the question that is still open after two centuries. This is part of the series: How X Solved Y. We cover: • the doubling test: 2p + 1, and why 5 survives but 7 (→ 15 = 3·5) does not • the Sophie Germain primes 2, 3, 5, 11, 23, 29, … and their safe primes 5, 7, 11, 23, 47, … • the Cunningham chain 2 → 5 → 11 → 23 → 47, and why it breaks (2·47+1 = 95 = 5·19) • why most primes fail: doubling a prime that is 1 (mod 3) lands on a multiple of 3 • Sophie Germain, Paris 1776–1831, M. Le Blanc, and her letters to Gauss • Germain's theorem: Case 1 of Fermat's Last Theorem holds for every Sophie Germain prime exponent • safe primes in cryptography: q − 1 = 2p keeps a large prime factor, defeating Pollard's p − 1 • the count up to 10^k vs the Hardy–Littlewood estimate 2·C₂·n / (ln n)² — and why infinitude is still unproven Every number on screen is computed, not estimated: the doubling map p → 2p+1 with its pass/fail factorizations, the Sophie Germain primes and their safe primes, the Cunningham chain and its break, and the real counts up to a million compared against the Hardy–Littlewood prediction. The history is sourced: Germain worked under the alias Antoine-August Le Blanc, corresponded with Lagrange and Gauss, and her result on Fermat's Last Theorem was attributed to her by Legendre in 1823. One question in the same corner is still open today: whether there are infinitely many Sophie Germain primes. Subscribe — @euclideayt. ——— Music by Vincent Rubinetti Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/a... CHAPTERS 0:00 The Doubling Test 1:28 The Survivors 2:52 The Chain 4:11 Why It Breaks 5:53 Sophie Germain 7:08 Germain's Theorem 8:46 Why "Safe" 10:26 How Many Are There 11:54 Coda