Posts Tagged ‘Paul Stanford’

stanford-paul-2009-08For 2ⁿ – 1 to be prime we also need n itself to be prime, but that is not sufficient. For example, 2¹¹ – 1 is composite even though 11 is prime.  However, if you look at tables of Mersenne primes it is interesting to note that if you start with 2 and use that to make a new number 2ⁿ – 1 with n = 2 you get 3, then recycling the 3 you get 7, use n = 7 and you get 127, another prime! How long could this go on?

Let f(n) = 2ⁿ – 1.  The iterations you get, starting from 2, are f⁰(2) = 2, f¹(2) = 3, f²(2) = 7, f³(2) = 127, f⁴(2) = 1701411834604692317316873037158884105727.

It turns out these are all prime!  But what of the next one???  Well, it may be a very long time before any of us know. The largest value of n for which 2ⁿ – 1 is known to be prime is n = 57885161, and that after a concerted effort using volunteers from around the globe.  Not much chance of answering this one in our lifetimes, unless some really new idea arrives.Before you make a hasty conjecture (as has already been done), a cautionary piece of history is in order.  If you define a new function to iterate you get some other interesting numbers.  Let g(n) = n² – 2n + 2. Then the iterates are g⁰(3) = 3, g¹(3) = 5, g²(3) = 17, g³(3) = 257, g⁴(3) = 65537, and all of these are prime!  So, with forgivable excitement, the conjecture was made that all of these will be prime, especially as the next one, g⁵(2) = 4294967297, was much too large at the time for mere mortals to conceive of factoring with their bare hands.

None, that is, until Euler combined his genius with an impish disbelief in Fermat’s conjecture to discover that g⁵(2) = 4294967297 = 641 * 6700417.  And since then we have found many more composite Fermat numbers, and no further Fermat primes, leading to the complementary conjecture that all the rest are composite!  It seems that we never learn to be humble around these things…

It takes a larger number to be “forever beyond reach” these days.  Rather than the now puny 4294967297 we cower before f⁵(2) = 2¹⁷⁰¹⁴¹¹⁸³⁴⁶⁰⁴⁶⁹²³¹⁷³¹⁶⁸⁷³⁰³⁷¹⁵⁸⁸⁸⁴⁴¹⁰⁵⁷²⁷ – 1, and who can blame us?

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Yet another Vi Hart doodling video, this time with a Christmas theme!

In this video she echoes the point made by our own Dr. Paul Stanford that every number has fascinating properties if you just think hard enough about them.

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stanford-paul-2009-08One of our favorite guest lecturers, Dr. Paul Stanford, was recently recognized for his work teaching college algebra, applied calculus, matrices and vectors, calculus and linear algebra.

The Regents’ Outstanding Teaching Award, which Stanford received in the category of contingent faculty, carries a $15,000 stipend.   Nominees are selected through a rigorous campus-based process beginning with deans and department chairs, relying heavily on student and peer faculty evaluations within academic departments, and then progresses through various stages of evaluation up through the university, resulting in a recommendation from the campus president. The selection committee evaluates annual reviews, evidence of continuous improvement, commitment to high quality undergraduate education, and other factors.

We look forward to having Dr. Stanford share his great talents with the Metroplex Math Circle again in the 2009-2010 season.

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Dr. Paul Stanford departed from our recent lectures on applied mathematics and contest preparation to give our students a glimpse into the fascinating world of pure mathematics.  Dr. Stanford is particularly skilled at teaching deep ideas without the need to resort to complex algebra, I’ll attempt in this recap to do his lecture a small bit of justice.

Dr. Stanford began with one of the simplest concepts, points on a plane and arrows connecting those points.  Many of these arrows considered together and connecting a set of points yield all sorts of interesting directed graphs or “digraphs.”  This idea becomes more interesting when restrictions are imposed such as the rule that no two arrows can originate from the same point.  When a sequence of arrows loop back upon themselves they form a shape like a “hairy circle” from the title of the lecture.

Dr. Stanford further restricted the cases with the following rules:  no two arrows can connect to the same point and no symmetry is allowed.    These restrictions yielded only four possibilities including those with only a single point connecting back on itself and the possibility of having no points or arrows at all.

Dr. Stanford then proceeded to show how basic arithmetic could be carried out in each of these four systems by shifting along the chains of arrows and by considering the very special case of the arrow that points back to its own origin.

In the second half of the lecture, Dr. Stanford built upon this foundation as he introduced the Collatz Problem:

Think of a number.

If it is even, divide by two.

Otherwise, triple it and add one.


Does this always reach one?

Dr. Stanford lead the students through multiple examples using positive integers, some of which filled both whiteboards but eventually came back to one.  In fact Dr. Stanford told us that while this process always yields one for numbers from 1 to 2.7 \times 10^{16},  nobody has proven that it is true of all positive integers!  Dr. Stanford reminded the students that for mathematicians multiple examples are not “proof” and that numbers even as large as 23 thousand, trillion are not especially large to mathematicians.

However, when Dr. Stanford started the Collatz problem with negative integers, it was common that loops would appear.  Drawn on the board, the chain of calculations was strangely similar to the hairy circles from the previous lecture.  This lead to a discussion of how one could detect when loops appear in an algorithm.  At this point the lecture began to  bridge between pure mathematics and some real problems in computer science.

One method for detecting a loop would be to track and remember every point in the path but this would burden even the most powerful computer.  A more clever method is the “tortoise and the hare” strategy.  This involves sending two runners through the system, one moving twice as quickly as the other.

Dr. Stanford proved how this method would eventually prove the existence of a loop and how a second tortoise could confirm the exact point where the loop begins.

The two lectures formed an exciting and satisfying trip through the world of numbers.  If I have misrepresented or ignored pertinent points from the lecture please feel free to mention them in the comments section below.

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Happy October 23

What is so special about October “23”? Well, as Dr. Paul Stanford has proven in his lectures there are interesting facts about every number! Here is what he has to say about the number 23. If you have any other facts to add or if you see any mistakes in my transcription of Dr. Stanford’s notes please use the comments below.

23 is…

The largest number not the sum of distinct powers.

With 23 people in a room, odds are that two share a birthday (better than 50:50.)

Prime, smallest odd prime not a twin.

Woodall number. 23=3 \cdot 2^3-1

One of the only two numbers that need 9 cubes. (The other is 239.)


If negatives allowed, 23=2^3+2^3+2^3+(-1)^3

23=0 \cdot 0!+1 \cdot 1!+2 \cdot 2!+3 \cdot 3!

23 is the smallest number of rigid rods that brace a square.

First prime where 23rd roots of unity form cyclotomic integers without unique factorization.

Number of trees with eight nodes.

Factor of 2^{11}-1

2^{23}-1 is composite: 47 \cdot 178481

Sophie Germain prime: 2(23)+1 also prime.

Wedderburn-Etherington number.

The first pillar prime.

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Dr. Paul Stanford is one of our favorite speakers at Metroplex Math Circle, famous for his book of fascinating facts about all the numbers from 0 to 100. If you are curious about the title of his lecture you will have to attend to satisfy your curiosity! But to give you a better sense of Dr. Stanford, here is his biography in his own words:

Let’s see…

I was born in Hampshire, England, in the same part of the country as the book Watership Down is set (the story about rabbit communities).

I have a doctorate in math, mostly because I was having such a good time doing math I didn’t want to stop. My research was in recursive structures and iterative proof methods on such structures.

I worked for Texas Instruments for many years, both in England and in Dallas, doing software development and then serving on standardization boards, helping electronic companies to share schematics and other descriptions of integrated circuits with each other. I was a senior member of technical staff (SMTS) at TI, and the standard was the Electronic Design Interchange Format (EDIF). I gave tutorials on EDIF at conferences around the world on many occasions, and was featured on a video training series.

After that I ran my own company for some years, designing computer languages for other companies. The company was called Custom Computer Languages. I have taught in community colleges and at UT Dallas for many years. I also developed automatic testing procedures within the telecommunications industry.

That’s about it. I live with my wife and granddaughter, have an embarrassingly large library of books and CDs, and have, from time to time, been guilty of writing poetry.

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