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Thomas Egense
Works at Statsbiblioteket
Attended Århus Universitet
Lives in Aarhus
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Thomas Egense

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For  #caturday here is my siamese  (Anubis) finding new opportunities with a tree I cut down  by using it as a look-out point.
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And how we have a result for cubes similar to Lagrange's four-square theorem

I notice that the result of seven cubes needed is a little more than I would expect from the four squares to obtain the same, and yet there even is a few numbers where eight cubes is needed. 

This new theorem should be named Siksek's eight-cubes theorem.

While this theorem may have no practical use, I am still satisfied to know the truth.
Number theory is famous for having a lot of easily stated but hopelessly difficult open problems (e.g. Goldbach's conjecture, the twin prime conjecture, etc...).  But the last few decades have seen a remarkable amount of progress on many of these, some of which have been open for centuries.  Here is another example that just hit arXiv today:  every integer greater than 454 is a sum of at most 7 positive cubes.
Abstract: A long-standing conjecture states that every positive integer other than 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, 364, 420, 428, 454 is a sum of at most seven positive cubes. This was first observed by Dase and Jacobi in 1851 on the basis of extensive ...
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* Five Podcasts from BBC about the most important programming languages*

The 15 minutes duration of each podcast is perfect timed with the driving time to my workplace...
Programming Languages

As part of the BBC's Make it Digital Season, Aleks Krotoski presents a brief history of some of the most famous high-level programming languages.  Each of these easily digested programmes is only fifteen minutes long and is available online as a stream and as a podcast or MP3 file.

Aleks Krotoski explores the history of programming languages. The history of computing is dominated by the hardware; the race for speed and power has overshadowed how we've devised ways to instruct these machines to do useful tasks.

Listen here (15 min streams):

These programmes should be available worldwide without restriction. They are easiest to play on a computer (Flash) although they will work on iOS with a few extra clicks and on Android after the BBC media player is installed.

The Tower of Babel:

Podcast and MP3s:

BBC Make it Digital:

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Ṫḩṓḿ Ƒᴙᴑᶊᵵ's profile photoDarryl “Ustād” Barnes's profile photoAllan MacDonald's profile photo
use to program in fortran 77
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New shipment arrived

The one to the right in wrap (Seaweed 120cm*120cm) is sold to +Jacques Riget and is copy #3  and final print of this fractal.

The two to the left (Turtle and Octopus) are replacements of the two sold to +Marselisborg Gymnasium.

The two middle smaller fractals are on printed on glass and due to reflections very hard to catch sharp on camera.
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Thomas Egense

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The counterintuitive Borwein Integral

For an explanation  see:

(Found on Quora)
Daria V's profile photoMichal Canecky's profile photoJun Wang's profile photoRonak Parekh's profile photo
Well .499999999999 can be written as .5 which is 1/2
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Yet another utterly futile attempt to find a md5 fixpoint

During lunch discussion at my workplace we started wondering if there existed a fixpoint for the md5 cryptographic hash function.

That is a point such that md5('x')='x' where x is the 32-digit hexadecimal string representation.

To be completely truthful the md5 algorithm works on 512 bit values.  A 32-digit hexidecimal string is 128 bit and is first padded(with zeroes) to 512 bits before the md5 function is taken and
returns a 128 bit value which then is converted to a 32 digit hexidecimal string thus explaining my md5('x')='x' expression.

Some googling showed this is known as the "Kember Identity" and is an unsolved problem. Also a few programmers had tried to brute force it over the years with no luck (not surprisingly as you will see below). Even though I would never succeed I decided to look into the problem anyway.

Interestingly there is a simple mathematical argument that there is ~63.21% probability  for at least one fixpoint. This does not mean there is one of course! But it also reveals there is probably only a few, if any! The argument is based on the same principly that when you shuffle a deck of card, what is the probability that at least one card will end up in the same position in the deck. The probability actually converges to 1-1/e very fast as the number of the elements shufled increases. There is no guarantee that the md5 function is a shuffling though, there could exist different 32-digit hexidecimal strings x,y such that md5(x)=md5(y). But such collisions are extremtly rare and would not change the overall probability by much.

Examples of different x,y with same hashing value does exist , see etc. But this example is 512 bit and not 128 bit (32 hexidecimal). The md5 hash function is not considered safe anymore due to collision vulnerabilities , but this is completly irrelevant to my mission of finding a fix point.
Lets see how futile a brute force attempt will be:
There are 2^128= 32^16 ~ 3.4*10E38 different values.
My brute force program can do 3.5M hash/sec (For each CPU used)
This gives 3*10E24 years to try all combinations (Using 1 CPU).
Not a very uplifting result and now you know why I used the word futile.
So without a cryptographic breakthrough, humanity will never know if such a point indeed does exist.

And now back to my brute force attempt:
I made a small Java program that can be found on Github on the link below.
Thanks to +Toke Eskildsen  for doubling the performance using tricky bit manipulation instead of the standard Java methods as substring, equals etc.
The code is much harder to read now though,  nothing comes for free...
The program starts with a random 32 digit hexadecimal string (for each CPU) and then iterative computes md5(x) and check if it matches x. Using this endless chaining of md5 values there is no performance overhead from generating a a lot of random strings. Of course there is a (tiny!) probability that this would result in an endless loop as it eventually is bound to do! But cycle detection would slow
the program down and detecting a cycle having 1E10 elements would take way too many resources anyway. Also I restarted the program
regulary over the 2 weeks I had it running on 24 CPUs and this would have reset any loops.

Since I would not find a fix-point but still would like to program to output somthing, I logged the maximum prefix and suffix match for md('x')='x'

The maximum match I found was a 12 character suffix and a 12 character prefix match shown below:

prefix 12:
54db1011d76dc70a0a9df3ff3e0b390f -> 54db1011d76d137956603122ad86d762

suffix 12:
df12c1434cec7850a7900ce027af4b78 -> b2f6053087022898fe920ce027af4b78

Remember this used 24*2weeks = 0.9 year CPU time. If you run it and find a 13 character match or more, I would like to hear for you of course.
MD5FixPointSearch - A simple java program that attempts to find a fix point for the MD5 function. It will try find a maximum prefix/suffix match.
David Jao's profile photoDarryl “Ustād” Barnes's profile photo
The bitcoin network currently produces 2^58 hashes per second (SHA2 hashes instead of MD5, but the idea is the same). At this rate it would only take 3.8E13 years.

On the other hand it would surely improve on the prefix and suffix matches.
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Thomas Egense

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Help teach science to students, save the planet and get an octopus!

The awesome team behind TWDK ( have started a fundraising campaign to cover some of the basic costs involved in running the site such as research, artwork, writing expense etc.

I was contacted by the TWDK  team and asked if I wanted to help with this charity campaign and since it both involved teaching science and environmental awareness,  I instantly agreed.

Six of my  best fractals all resembling marine creatures can be picked as rewards when donating more than £150 GBP. I am selling the fractals very close to my production/shipping cost  so donaters will get some value for their money.  The total price for supporting the project and getting a fractal is still less  than my regular price for the same fractal.

The campaign is scheduled to run for 60 days.

+John Baez
Free climate change resources for schools | Crowdfunding is a democratic way to support the fundraising needs of your community. Make a contribution today!
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Game of thrones Season five to be premiered TODAY

And if you want a small recap of what happened in the previous four seasons these gorillas have a very short version of the plot  :)

via +Søren Sprogø 
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The Danish Newspaper Online Archive is now online

After 200+ years of collecting all danish newspapers and stacking them in warehouses or copying them to microfilms, they can  now   be  accessed online for free!  At The State And University Library Aarhus we have been working on this project for years and are very happy to open it to the public today.

We have build a powerful search engine on top of all data so it is easy to find text matches down to a specific page with  highlighting and you can download the newspaper as a PDF. For historians this is a gold mine and  the general public will be able to search for family members or old articles they want to read again.

So far we have only 1 million news paper pages in the index, but this will increase to over 32 million pages over the next year.

You can only search in newspapers that are older than 100 years, unless you search from the computers within the State and University Library where there is no limitations.

The OCR (Optical character recognition) is not perfect and this is most evident in oldest newspapers, but we have several ideas how to improve this over time.

Since we just went live today there is a heavy load on the site - so be gentle :)

+Peter Mouritsen
+Toke Eskildsen
+Mads Villadsen
+Per Møldrup-Dalum 
Din e-mail-adresse (hvis du vil have svar fra os): OK Mediestream bruger cookies til at lave statistik over trafikken på siden og til at forbedre brugeroplevelsen. Ved at klikke videre accepterer du brugen af cookies, som du i øvrigt til enhver tid kan slette. Læs mere om vores cookiepolitik.
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Robert Langlands  - the mathematician behind the 'Langlands Program'.

The 'Langlands program' can be compared to the similar hunt for a "Unified Theory" in physics. The program is trying to connect previous believed totally unrelated branches of mathematics and some of the found connections is still a mystery.

If you want a much deeper understanding about this project I can recommend reading "Love and Math" by Edward Frenkel. But the book require some mathematical knowledge to understand the concepts in more details. I did not understand everything since it require knowledge in so many different branches of mathematics.  Still the book is the best way to gain a deeper insight into the frontiers of mathematical research.
Canadian Robert Langlands is 'like a modern-day Einstein,' who has devoted his life to the limits of pure mathematics
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Lige så meget som muligt 🐘
 ·  Translate
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The counterintuitive Borwein Integral

For an explanation  see:

(Found on Quora)

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Funny 😂
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Article in The New Yorker about Yitang Zhang, this time with more personal information about himself and his life, and not only about his mathematical breakthrough.
Excellent piece in The New Yorker about Yitang Zhang and his groundbreaking work establishing that there is a bound on the gap between two consequent primes no matter how far along we are on the road to infinity.

Zhang's example shows that it's possible to do top level mathematical research outside of the academia. Good new for all of us, and... shame on you, academia!
Unable to get an academic position, Zhang kept the books for a Subway franchise. Credit Photograph by Peter Bohler
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I've always thought along these lines. Science is for everyone
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  • Århus Universitet
    Cand. Scient, Mathematics/Physics, 1991 - 2000
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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.
Keywords that describe me as a person: scientist, mathematician, sceptic, atheist, blogger, and digital artist.

I blog about my mathematical adventures, where I, among other things, solve various puzzles,  which often involves brute force CPU attack on the problems.

In my spare time I use mathematics to create art.

For the last 12 years I have worked as a Java programmer.

I like retro-gaming, and I never become tired of talking about old-school games or computer
music (C64/Amiga).

I love animals - especially cats - and I have two cats of my own.
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Cream of the Crop 15/4-2012:
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    IT Consultant, 2011 - present
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    Software developer, 2005 - 2006
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Thomas Egense's +1's are the things they like, agree with, or want to recommend.
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