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answered Is $p_n \sim \frac{5}{4}n\log(n) + \frac{1}{2}n + \frac{(p_1+\ldots+p_{n-1})}{n-1}$ a good approximation for the $n^\text{th}$ prime?
Aug
28
comment Is there a number which describes the approaching ratio of twin primes to other primes? Or a formula for the change in density of twin primes?
@gebra: It should look more and more like a line the further you zoom out, since the log 4/log x term approaches 0 from above.
Aug
28
comment can Sophie Germain prime be arbitrarily many?
@iadvd: Well, go through it and skip what you don't understand. You might still catch a logic error.
Aug
28
answered $O( n^3)$ vs $O(n^2 \ log n)$
Aug
28
comment can Sophie Germain prime be arbitrarily many?
@iadvd: The paper has an error on page 6, and generally seems to be at the wrong level for this sort of discovery. Flipping through the paper I see that the proof on pp. 32-33 is also flawed, so I imagine its main result is likewise not correct. If you're interested it might be a good exercise to work through their proof and see if you can discover mistakes.
Aug
27
comment Is there a number which describes the approaching ratio of twin primes to other primes? Or a formula for the change in density of twin primes?
@gebra: The graph is reversed, with total primes on y and twins on x. As you double the value on the x axis from x to 2x you expect the value on the y axis to increase from y to about (2 + log 4/log x)*y.
Aug
27
comment Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
@DanChristensen: Depends on your definition of aleph numbers, I suppose. You could replace omex above with aleph0 if you prefer.
Aug
27
comment Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
@goblin: You can examine the proof to see the nitty-gritty details if you like, but they already need essentially all this machinery to prove the Peano axiom that any set containing zero and closed under the successor operation contains all natural numbers.
Aug
27
answered Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
Aug
26
answered Is there a number which describes the approaching ratio of twin primes to other primes? Or a formula for the change in density of twin primes?
Aug
24
comment Besides 1 and 11, is $\sum_{i=0}^n 10^i$ composite for every $n\in \mathbb{N}$?
@Omnomnomnom: Dubner says that the only viable way to prove primality for a number of this size is the BLS test, but we'd need to know more of the factorization of 10^49080-1 before we could do this -- we need ~33% but we have only ~16% now.
Aug
17
comment Does the sum of reciprocals of primes congruent to $1 \mod{4}$ diverge?
@RaceBannon: As I said, it's pretty easy to prove given the PNT in APs. This even gives the correct asymptotic -- $$\sum_{p\le x,\ p\equiv k\pmod m}\frac1p\approx\frac{\log\log x}{\varphi(m)}$$.
Aug
17
comment Does the sum of reciprocals of primes congruent to $1 \mod{4}$ diverge?
@RaceBannon: It's easy given the Prime Number Theorem in arithmetic progressions -- or did you just mean that the PNT is hard to prove?
Aug
17
reviewed Approve Does the sum of reciprocals of primes congruent to $1 \mod{4}$ diverge?
Aug
17
answered Does the sum of reciprocals of primes congruent to $1 \mod{4}$ diverge?
Aug
14
answered Does the smallest prime factor of a Fibonacci number appear in the Fibonacci sequence?
Aug
14
reviewed Leave Closed How To Prove Irrational Square Roots and Inequalities In Courant's Calculus Book?
Aug
14
reviewed Close What is mean by basis of a vector space?
Aug
14
reviewed Close difference between $\mathbb{A}^n$ and $k^n$
Aug
14
reviewed Leave Open Critique of this subgroup proof?