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Thomas Egense
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I have a math blog and I am generally interested in all kinds of scientific matters. I use mathematics to create art – you can check out my album. I try to find unique content for my posts,which are mostly of scientific nature.
I have a math blog and I am generally interested in all kinds of scientific matters. I use mathematics to create art – you can check out my album. I try to find unique content for my posts,which are mostly of scientific nature.

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Curious pattern in most common gap between prime numbers.
Mind the gap!

People love to talk about twin primes. But a computer search has shown that among numbers less than a trillion, most common distance between successive primes is not 2 but 6. This goes on for quite a while longer…

… but Andrew Odlyzko, Michael Rubinstein and Marek Wolf have persuaded most experts that somewhere around x = 1.7427 ⋅ 10³⁵, the most common gap between consecutive primes less than x switches from 6 to 30:

• Andrew Odlyzko, Michael Rubinstein, and Marek Wolf, Jumping champions, Experimental Mathematics 8 (1999), 107–118. Available at http://projecteuclid.org/euclid.em/1047477055

This is a nice example of how you may need to explore very large numbers to understand the true behavior of primes.

They give a sophisticated heuristic argument for their claim — not a rigorous proof. But they also checked the basic idea using Maple’s ‘probable prime’ function.

It takes work to check if a number is prime, but there’s a much faster way to check if it’s probably prime in a certain sense. Using this, they worked out the gaps between probable primes from 10³⁰ and 10³⁰+10⁷. They found that there are 5278 gaps of size 6 and just 5060 of size 30. They also worked out the gaps between probable primes from 10⁴⁰ and 10⁴⁰+10⁷ There were 3120 of size 6 and 3209 of size 30.

So, it seems that somewhere between 10³⁰ and 10⁴⁰, the number 30 replaces 6 as the most probable gap between successive primes! But their heuristic argument more precisely pins down the location where it switches.

Using the same heuristic argument, they argue that somewhere around 10⁴⁵⁰ , the number 30 ceases to be the most probable gap. The number 210 replaces 30 as the champion—and reigns for an even longer time.

Furthermore, they argue that this pattern continues forever, with the main champions being the ‘primorials’:

2

2⋅3 = 6

2⋅3⋅5 = 30

2⋅3⋅5⋅7=210

2⋅3⋅5⋅7⋅11=2310

and so on.

#bigness
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Today's formula
The Borwein Integral is counterintuitive!

+Greg Egan has extended the Borwein Integration principle such that it holds for
n<1.5*10E43.
For more info see the post by +John Baez.

And thanks to +Per Møldrup-Dalum for the nice gif.
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After 6 years has the 'proof' of the ABC-conjecture by Shinichi Mochizuki finally been proved flawed? It indeed seems so!

But first a small summery of the last 6 years!

In August 2012 the japanese mathematician Shinichi Mochizuki sent shockwaves through the global mathematics community by claiming he had a proof for the ABC-conjecture. (https://en.wikipedia.org/wiki/Abc_conjecture)

A proof for the ABC-conjecture would have profound impact on mathematics because this would also solve a lot of other unknown problems. Even Andrew Wiles proof of Fermat's Last Theorem would just follow from the ABC-conjecture.

So the news was received quite sceptical by the mathematics community. It was simply too good to be true even though Shinichi Mochizuki was a well known respected mathematician. However just one year later Yitang Zhang made a major breakthough on the Twin Prime conjecture, a problem that had also eluded mathematicans for centuries. So breakthroughs in mathematics of this magnitude was possible. And in this case the proof was verified very fast by various mathematicians. But what about the proof of the ABC-conjecture?

The scepticsm was followed by dispair. The paper was hundred of pages and noone could understand it. It was described by other mathematicians as it almost was written in an alien language called "Inter-universal Teichmuller Theory"...

Definitions after definitions and very simple implications keep going on for page after page and in the end the ABC-conjecture has been proved. Famous mathematician Terence Tao from UCLA noticed that it was peculiar that the details in the proof of such a strong theorem revealed no other deep insights into mathematics. There was no smaller building blocks where each block in itself would reveal something new about mathematics.

The mathematics community did not give up on the proof and the Polymath project on the ABC-conjecture was founded. (http://michaelnielsen.org/polymath1/index.php?title=ABC_conjecture). But even with a strong team of brilliant mathematicians they were going nowhere on the proiect attempting to crack the proof. Month after month passed... Rumours say Shinichi Mochizuki himself was very reluctant in helping the Polymath project by answering their questions. I guess it was a very frustrating workproject for the participants... News from the projects became rare and all their work seemed to be in vain.

Fast forward to this week!
Two mathematician claims to have found a flaw in the proof and they have strong case.
The two mathematics behind the claim:
Peter Scholze: Fields Medalist winner 2018 and known for his ability to crunch hard math books in with the same speed as normal people are reading comics.
Jakob Stix: Expert in Anabelian Geometry, a field in mathematics as close as you can get to Shinichi Mochizuki's work.

And what did these brilliant mathematicians find? Supposedly a flaw in a symmetry argument that is crucial for the proof. They showed the symmetry was broken by 'going around in the other direction'. I have no illusions to further grasp any details in their argument except this statement:

"But when you go around the circle, Stix said, you end up with a measuring stick that looks different from if you had gone around the other way."







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50th Mersenne Prime Discovered
2^77232917 - 1 is prime

It was discovered on December 26, 2017 by Jonathan Pace, and verified to be a prime by GIMPS using four different programs on four different hardware configurations.
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Breakthrough Prize in Mathematics 2017 awarded
They proved the number (different shapes) of solutions for Diophantine polynomial Equations over the complex numbers is always finite.


Originally shared by ****
Breakthrough Prize in mathematics: Christopher Hacon [1] and James McKernan [2].

New Horizons in Mathematics Prizes Awarded to Aaron Naber [3], Maryna Viazovska [4], Zhiwei Yun [5], and Wei Zhang [6].

[1] https://en.wikipedia.org/wiki/Christopher_Hacon
[2] https://en.wikipedia.org/wiki/James_McKernan
[3] http://www.math.northwestern.edu/~anaber/
[4] https://en.wikipedia.org/wiki/Maryna_Viazovska
[5] https://en.wikipedia.org/wiki/Zhiwei_Yun
[6] https://en.wikipedia.org/wiki/Wei_Zhang_(mathematician)
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Dynamic graph of HTML tags usages from 2005-2017 in the Danish Netarchive

See the relative frequency of HTML-tags over time on webpages. What happened to the hated blink-tag? How fast was canvas-tag embraced by the community? See for yourself!

The tags registered are not limited to legal HTML tag, but anything that is marked on a tag on a HTML page. Try sarcasm-tag etc.

The Danish Netarchive consists of 10.5 billion harvested HTML pages from 2005 up to today. This amounts to 2.4 Petabyte of archived data.

Tags Demo: http://labs.statsbiblioteket.dk/tags/

(Labs main page: http://www.statsbiblioteket.dk/sblabs/)
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The Windshield Phenomenon
Troubling news about a 80% decrease of the biomass of insects.

http://www.sciencemag.org/news/2017/05/where-have-all-insects-gone

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One of John Conway's open math problems solved.
The famous mathematician John Horton Conway has a list of 5 open problems and one of the problems now has been solved. And it is one that is easy to understand for non-mathematicians as well.

Numberphile just did a video with the solution:
https://www.youtube.com/watch?v=3IMAUm2WY70

Any cycle would have been a counter example, but James David actually found a fix-point - a cycle of length 1.

It reminds me of the Collatz' conjecture. Mathematicians has no clue how to maky any progress on the conjecture. Best hope is someone finding a counter example.

John Conway's 5 open problems:
https://oeis.org/A248380/a248380.pdf

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