Mathematician Dan Brown Proves a Prime-Prefix-Free Sum Converges, Assuming the Riemann Hypothesis
The reciprocal sum of the prime-prefix-free numbers converges [pdf]
Dan Brown, author of The Da Vinci Code, proves that the sum of reciprocals of prime-prefix-free integers converges under the Riemann Hypothesis. The proof combines Selberg's theorem on primes in short intervals with an upper-bound sieve for prime pairs, and is machine-verified in Lean 4. The sum exceeds 3.5276 unconditionally and is bounded by about 5×10^14 under RH. Convergence hinges on 1/log 2 ≈ 1.4427 > 1, meaning base 2 is the only base where the analogous sum converges.
Convergence therefore hinges on 1/log 2 ≈ 1.4427 exceeding 1. In base b the same model gives the exponent 1/log b, below 1 for every b ≥ 3: binary is the one base in which the analogous sum is expected to converge.
- jdb1729
The reciprocal sum of the prime-prefix-free numbers (https://oeis.org/A287117) converges to a number less than 5*10^14, conditional on the Riemann Hypothesis.
This Lean-verified proof answers a question I posed 10 years ago: https://math.stackexchange.com/questions/2288648/does-the-su...
An equivalent version: if we start with 1 and then output a stream of random bits, reading the number as a big-endian binary number at each step (so each time a bit arrives, the number is multiplied by 2 and 1 is either added or not), the expected time until the number is an odd prime is finite.
- dash2
Why does it matter to hn? Is it because Dan Brown wrote it?
- yzydserd
“Author of The Da Vinci Code”
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