Intereting Posts

Parity through Series expansion
Prove any function $f$ is Riemann integrable if it is bounded and continuous except finite number of points
Contour integration of $\frac{\log( x)}{x^2+a^2}$
Recovering eigenvectors from SVD
study of subspace generated by $f_k(x)=f(x+k)$ with f continuous, bounded..
minimize a function using AM-GM inequality
Are there examples of when the ILATE mnemonic for choosing factors when integrating by parts fails?
Show that $a^3+b^5=7^{7^{7^7}}$ has no solutions with $a,b\in \mathbb Z.$
$\frac{1}{{1 + {\left\| A \right\|} }} \le {\left\| {{{(I – A)}^{ – 1}}} \right\|}$
On the Paris constant and $\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}$?
Series of logarithms $\sum\limits_{k=1}^\infty \ln(k)$ (Ramanujan summation?)
Conditions for the equivalence of $\mathbf A^T \mathbf A \mathbf x = \mathbf A^T \mathbf b$ and $\mathbf A \mathbf x = \mathbf b$
How do I go about proving da db/a^(-2) is a left Haar measure on the affine group?
Number of copper atoms in $1\mbox{cm}^3$ of copper
Is $$ the union of $2^{\aleph_0}$ perfect sets which are pairwise disjoint?

This is essentially a Bertrand’s postulate version for twin primes. I am interested in both an explicit example and large lower bounds for it because of this answer of mine. In the comments below the answer, it is shown that there is no such $n$ below $8\times 10^{15}$.

An efficient algorithm would be as follows: take an initial point $m$ for which Bertrand’s postulate for twin primes is true (say, $13$). Find the greatest prime twin $p\lt 2n$. The new initial point is $p$. Iterate.

An explicit example of such $n$ would cause a very large gap $\approx n$. Although it seems quite unlikely for such $n$ to exist, a proof remains far from reality, so I am interested in a computational effort.

- Proving an expression is composite
- Find a positive integer such that half of it is a square,a third of it is a cube,and a fifth of it is a fifth power
- Proof of Cohn's Irreducibility Criterion
- Variation of Pythagorean triplets: $x^2+y^2 = z^3$
- Cramér's Model - “The Prime Numbers and Their Distribution” - Part 2
- Multiplication by $m$ isogenies of elliptic curves in characteristic $p$

Do we know $n\gt 5$ with no twin prime $n\lt q \lt 2n$? If not, what’s the best known lower bound?

- Prove or disprove: if $x$ and $y$ are representable as the sum of three squares, then so is $xy$.
- If $n = a^2 + b^2 + c^2$ for positive integers $a$, $b$,$c$, show that there exist positive integers $x$, $y$, $z$ such that $n^2 = x^2 + y^2 + z^2$.
- Can an odd perfect number be divisible by $101$?
- Is the Euler phi function bounded below?
- Nature and number of solutions to $xy=x+y$
- If $a \mid m$ and $(a + 1) \mid m$, prove $a(a + 1) | m$.
- Detecting perfect squares faster than by extracting square root
- General formula to obtain triangular-square numbers
- Prove that $x^2+1$ cannot be a perfect square for any positive integer x?
- Please verify my proof of: There is no integer $\geq2$ sum of squares of whose digits equal the integer itself.

We can’t yet prove that there are infinitely many twin primes, so we certainly can’t prove that there’s a twin prime in (n, 2n). But it’s surely true. Indeed, between n and 2n we expect there to be about $Cn/\log^2 n$ twin primes for some positive constant $C$.

If you look at the worst case, consider A113275: Kourbatov found that the twin prime 1121784847637957 was followed by a longer gap of non-twin primes than any smaller twin prime, but the distance to the next twin prime was just 23382. That’s a lot smaller than the 1121784847637957 you’d need to make the twin prime Bertrand fail!

Using the same approach as Michael Stocker I was able to check that there are no exceptions up to $10^{120}.$

*Edit:* I extended the range to $10^{262}$ and proved all the primalities. Time required was a few hours.

You can get very far with very little effort.

I think an example to make your algorithm clearer wouldn’t be bad.

If we have a twin-prime p,p+2, for every n in the interval [((p+2)+1)/2 , p-1] there is (at least) one twin-prime in [n,2n], namely p,p+2.

We know that 17 and 19 form a twin prime. So every n in [10,16] satifies “Bertrands postulate for twin-primes”

So the next twin-prime we want so find should generate a Interval that starts no earlier than 17. So we want ((p+2)+1)/2 < 18. Now we look for the biggest twin prime that satisfies that unequality.

We find the twin-prime 29,31 and the Interval [16,28] —

41,43 ; [22,40] —

71,73 ; [37,70] —

137,139; [70,136] —

……

I let my pc compute for only a few minutes without much optimization and got to the 96 digit twinprim

65401729995203484466533471060061363152550275078004047522007054662124597261449032607389217013721 (+2)

- Proof by induction that $1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}$
- Determine if the sequence is convergent or divergent
- Center-commutator duality
- $QR$ decomposition of rectangular block matrix
- What is more elementary than: Introduction to Stochastic Processes by Lawler
- For which topological spaces $X$ can one write $X \approx Y \times Y$? Is $Y$ unique?
- Find the follwing limit if $f$ is differentiable
- Proving divisibility of expression using induction
- Finding the distance between a point and a parabola with different methods
- How to prove $b=c$ if $ab=ac$ (cancellation law in groups)?
- Is the function $ f(x,y)=xy/(x^{2}+y^{2})$ where f(0,0) is defined to be 0 continuous?
- How to find the quotient group $Z_{1023}^*/\langle 2\rangle$?
- Partial derivative of a sum with several subscripts
- Discrete math induction problem.
- Uniform convergence of functions, Spring 2002