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1h
comment Square in Interval of Primes
The last element in the sample sequence 2,5,10,... should be 77 not 75.
6h
comment Square in Interval of Primes
possible duplicate of Existence of perfect square between the sum of the first $n$ and $n + 1$ prime numbers
Sep
11
comment Renyi entropy of prime gaps
I did not see this question until today. Your $S$ and mine appear to be almost the same (I do not square $H$ in the denominator of $f$). While I have an estimate it is only a guess based on a few calculations. Now that I see our sums are not identical maybe it's not useful to you. math.stackexchange.com/questions/893875/sorting-of-prime-gaps.
Aug
15
comment Closed form of $\int_0^2\frac{1}{2+\sqrt{3\,e^x+3\,e^{-x}-2}}dx$
Mathematica gives $ \int f(x)dx = \int \frac{dx}{2+\sqrt{6\cosh x-2}} = I(x) = (1/6) \left[(2 - \sqrt{-2 + 6 \cosh x}~)\cdot \coth \frac{x}{2} - 2~ i~ E_2((i~ x/2),~ 3) - 4~i ~E_1((i~x/2),~ 3)\right]. $ Using Santosh Linkha's idea, and taking care to take the $\lim_{a\to 0} \int_a^2$ gives the answer in the OP. $E_1,E_2$ are elliptic integrals.
Jul
24
comment Limit of Sum $\sum \frac{p(k+1) - p(k)}{p(k)^2}$ ? (corrected)
Yes, thanks, I don't trust my results that far but the differences were getting small quickly.
Jul
20
comment Euclid's proof of the infinitude of primes to prove this question
@MichaelHardy: This is a really nice paper and worth several readings. Thanks for the reference and the surprising answer.
Jul
19
comment Euclid's proof of the infinitude of primes to prove this question
@MichaelHardy: Your comment surprised me. I understand that $S$ is then extended to a larger finite set of primes. If it is error to call it a proof by contradiction it is a common error. Would you mind elaborating briefly?
Jul
9
comment Are NSA Mathematicians second-rate?
This doesn't seem to be about math.
Jul
8
comment Asymptotic density of powers of primes
Well--not so sure. $\pi(11) = 5.$ But we have $2,2^2,2^3,3,3^2,5,7,11$ or cardinality of 8...?
Jul
8
comment Asymptotic density of powers of primes
That's how I understand it although I am not familiar with $\Pi(x)$ in that form. It is the cardinality of the set of powers of primes less than or equal to x. At least that's how it looks to me.
Jul
8
comment Asymptotic density of powers of primes
Under the Wiki entry for Prime Number Theorem, Chebyshev's $\psi(x) = \sum_{p^k \leq x} \log p$ is asymptotic to x. Comparing this to your $\Pi(x)$ would seem to confirm Antonio Vargas' answer and I think give a proof. You should write $\psi(x)$ as a product.
May
26
comment Statements with rare counter-examples
This rings a bell...duplicate?
May
13
comment A product for 1/e?
Do you think for $|F_i|< 1|$ we could also just say $\log(1 + (-F_i)) \approx -F_i,~$ ...etc.? And so $\sum (-F_i) \approx -1$ and exactly in the limit as n gets large?
May
12
comment A product for 1/e?
@Ian: Fair enough. I only have time to remove it at the moment. I think my computer was using a $\Gamma$ function or something to interpolate and I will have to go back and check.
May
11
comment Grade School Math: Bad math, or new meanings?
@DavidMitra: A digit represents one of ten numbers. But if I stipulate that it the digit 1 also represents (say) 1000 by virtue of its place in a number that might be a contract if you understand and accept it. The problem with the question as I see it is that this is not at all clear.
May
11
comment Grade School Math: Bad math, or new meanings?
Right, I see both sides but it's not a good test question. Additionally, if the answer is (1), what does "1 times greater than..." mean?
May
1
comment sum of primes: approximate closed form?
If you had an approximate expression for the percentage of primes from 1 to n and multiplied it by n that would give you an approximate expression for the number of primes from 1 to n...
Apr
24
comment Pseudo Proofs that are intuitively reasonable
@robjohn: It is talking about the same function, yes, and I think my argument is overstated.
Apr
24
comment Pseudo Proofs that are intuitively reasonable
@robjohn: I think my answer is essentially a duplicate of yours. If you agree I will take mine down in the interest of good site husbandry. I didn't see this at the time.
Apr
22
comment Pseudo Proofs that are intuitively reasonable
@robjohn: The idea was paraphrased from an unnamed source in the text I cited. I will look at your answer (time permitting) and edit mine if that seems indicated. The answer was not given with reference to yours and I don't doubt your assertion is correct. Thanks.