The Mathematics of Paul Erdös II


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Expertise. Insights. Illumination.

Actually, I introduced him on the stage of the National Academy of Sciences when he won a mathematics prize some years ago and I used a juggling ball to do it, since his work involved sphere packing. If Noam said he did it, then I believe it! The year-old Dr. Why did it take Dr. Elkies to solve this problem? Was the problem that difficult?

The Mathematics of Paul Erdös II | Ronald L. Graham | Springer

I sent an e-mail to David desJardins and received this reply:. The main insight here is that some of these configurations are inconsistent, e. This is a small enough number to exhaust over, so the rest is pretty easy. I followed up with Dr. Elkies as well: what did he think when he saw the problem? And what were the keys to solving it? I was surprised that such a question could still be open. My second thought was that 8 is small but the number 56 of triples is still too large: as David desJ. But it was not obvious whether this would suffice to bring the computation down from the extravagant 2 56 to something feasible.

This is the start of the famous sequence of Catalan numbers, and it turns out to be a nice exercise to prove that this pattern persists for all n. This already makes the problem feasible: remember a computer can do elementary operations in a fraction of a second, and even though testing a given collection of triples is not elementary I figured it would be straightforward enough to check them all within an hour, and then do linear algebra to screen the others and either prove or disprove the conjecture. First I figured I might as well impose the full consistency condition and now we re-join the path that David DesJ.

The next power of 2 in this pattern is 2 21 or about 2.

Erdos Discrepancy Conjecture

There were further refinements available, such as exploiting the symmetry between say 0,1,2,4 and -4,-2,-1,0, which are clearly equivalent but have different AP3 patterns; this might roughly halve the computing time, but the extra effort to code this correctly would not be worth the time saving. But by now I already had all the pieces to finish the computation.

Paul Erdős

This took less than 20 seconds on my laptop. Fortunately modern computers can also do linear algebra problems of this size faster than we can set up the computation. Having gone this far I thought it might be interesting to check for configurations that can be colored in only one way, and found the two that I posted on your blog plus their mirror images. This still leaves the power-of-2 mystery. That could be much easier to prove than the power-of-two formulas for the numeration of all consistent configurations.

Thank you, Dr.

Elkies, and for everyone who participated this week. This level of detail can be found throughout the book, creating an enduring adventure for kids of any age: from 3 to 3 4 and beyond. It is the belief that total disorder is impossible. A problem in Ramsey Theory is what is the smallest set that contains a certain object. For example: What is the smallest group of people that contains two people of the same sex?

The answer is three. The classic example is the "party problem". What is the smallest party which will contain a certain number of people who know each other or who are all strangers?

See a Problem?

For three people the answer is six. Consider a party of 6. Let one member of the party be named Brent. Since Brent either knows or doesn't each person he will know at least 3 people or not know at least 3. The world has lost one of the greatest, most prolific, most original and most loveworthy mathematicians of all time.

I shall try to elucidate on all four adjectives of the previous sentence because they are all meant in al earnesty. He started as a number theorist and some proofs from his early youth are still amazing to this day for their exceptional mathematical beauty.

Mathematics Genealogy Project

He also became one of the top authorities in classical set theory and founded quite new branches of it, like e. But he also founded combinatorial geometry, transfinite combinatorics, and many more often with co-authors see below.


  • Paul (Pál) Erdős.
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He has literally thousands of theorems which count as key results in their respective disciplines. The number of his published papers is around and another 50 or more are still to be published after the death of their author.

Solution: ‘Darwinian Evolution Explains Lamarckism’

Two people A. Hajnal and A. But he was exceptionally prolific in other ways too. As it is well-known he travelled widely and unceasingly, and wherever he went he gave talks. These were not only the usual scholarly type of talks about new and earlier results of himself and others he very often gave talks on unsolved problems in geometry, number theory, combinatorics, etc. He had a few more favourite topics, among them "The problems I would most like to see solved" and "Child prodigies".

He also wrote quite a few books, usually with co-authors.

The Mathematics of Paul Erdös II The Mathematics of Paul Erdös II
The Mathematics of Paul Erdös II The Mathematics of Paul Erdös II
The Mathematics of Paul Erdös II The Mathematics of Paul Erdös II
The Mathematics of Paul Erdös II The Mathematics of Paul Erdös II
The Mathematics of Paul Erdös II The Mathematics of Paul Erdös II

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