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.
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.
Yes, see the table in Remark 7.3 on page 5, it exceeds 3.5, with growth slowing to a crawl. But the calculations mean little, sum(1/p) grows as divergent log(log(n)), so it also has the appearance of convergence on that basis. Many on math.SE argued for divergence (answers since deleted)! Although the proved upper bound is 5e14, heuristically it should be less than 4. I doubt RH is truly necessary. But even relying on RH, the exact value of the sum is elusive.
Well, not really, the contrapositive is that if the series diverges the Riemann Hypothesis would be refuted. But few doubt that the Riemann Hypothesis is true. So using it as an assumption merely makes the convergence slightly iffy. For another example of its use, see the deterministic Miller primality test: https://en.wikipedia.org/wiki/Miller–Rabin_primality_test
I don't know, what if someone made an app or a website where you get a dollar for every bit until an odd prime hits. The theorem says how much to price each spin to guarantee a long-term profit for the house (somewhere between tree fiddy and half a quadrillion dollars).
Actually, nothing practical, just sharing my love of math and excitement about the new possibilities of formal verification being unlocked.
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.
Just for reference, the sum of all primes is infinite https://en.wikipedia.org/wiki/Divergence_of_the_sum_of_the_r... so this result is not obvious.
Anyway, I think it's weird it depends on the Riemann Hypothesis.
Do you have some numerical test for intervals like sum up to 1000, up to 10000, up to 100000, up to 1000000, ... ?
Yes, see the table in Remark 7.3 on page 5, it exceeds 3.5, with growth slowing to a crawl. But the calculations mean little, sum(1/p) grows as divergent log(log(n)), so it also has the appearance of convergence on that basis. Many on math.SE argued for divergence (answers since deleted)! Although the proved upper bound is 5e14, heuristically it should be less than 4. I doubt RH is truly necessary. But even relying on RH, the exact value of the sum is elusive.
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that's a result that says more about the Riemann hypothesis than this specific problem right?
Well, not really, the contrapositive is that if the series diverges the Riemann Hypothesis would be refuted. But few doubt that the Riemann Hypothesis is true. So using it as an assumption merely makes the convergence slightly iffy. For another example of its use, see the deterministic Miller primality test: https://en.wikipedia.org/wiki/Miller–Rabin_primality_test
Ok? Why is this significant?
I don't know, what if someone made an app or a website where you get a dollar for every bit until an odd prime hits. The theorem says how much to price each spin to guarantee a long-term profit for the house (somewhere between tree fiddy and half a quadrillion dollars).
Actually, nothing practical, just sharing my love of math and excitement about the new possibilities of formal verification being unlocked.
“Author of The Da Vinci Code”
?
That would be Leonardo di ser Piero da Vinci
A nom de OOM.
out of memory?
Why does it matter to hn? Is it because Dan Brown wrote it?
Because the proof and Lean formalization have been produced by a clanker.