More Vi Hart doodles, this time exploring number theory.
Posts Tagged ‘number theory’
Number Theory is one of the cornerstones of math competitions and a vast field explored by professional mathematicians. We are very fortunate to have Dr. Josh Nichols-Barrer present to us next week. Dr. Nichols-Barrer was a very accomplished teenage competitive problem solver who has earned his doctorate at MIT. In addition to Number Theory our students can learn a great deal from his personal experiences. Here is a description of the session in Dr. Nichols-Barrer’s own words:
As you know, each positive integer may be factored uniquely into a product of powers of primes. Do you know why? In this class we will look at the structure of the integers in the simplest terms, and use that for a foundation from which we might actually prove that what we know to be true actually is. In the first half we will recall some things about arithmetic that we know intuitively and can name (with maybe a surprise or two), and in the second half we will proceed to prove (along with some other things which are no less key) unique prime factorization in the integers.
Our September 18th speaker, Dr. Branislav Kisacanin, is an accomplished author. His book, Mathematical Problems and Proofs, has enjoyed very favorable reviews on Amazon:
This is a very interesting book. If you have mastered the bare essentials of set theory (through the upside down A for “for all” and the backwards E for “there exists”), AND either 1) the bare essentials of combinatorics (through Pascal’s triangle), or 2) the bare essentials of number theory (through the definition of the Moebius function and the statement of the Chinese Remainder theorem), or 3) the bare essentials of geometry (through the law of cosines), AND if you are very talented in mathematics, then this book is a “MUST READ”. It matters not whether you are a high school student or a professional mathematician. You will find new and fruitful insights and quite a few interesting problems in this book. For the beginner, there are several tantalizing (but somewhat oversimplified) references to advanced topics such as Paul Cohen’s proof of the independence of the continuum hypotheses and Wiles’ proof of Fermat’s last theorem. For the professional there are footnotes with references to little known and suprising results obtained in the 20th century. But, unlike the claims in the “editorial review”, this book neither prepares you to read the literaure nor is it a store house of exercises which will help you take your problem solving abilities to the next level. The “editorial review” is “off”, but the book is “right on”. Only the title is unfortunate. Where it will help the career mathematician is in the “beer hall” or “coffee house” or “tea time” discussions with other mathematicians. It is just chock full of beautiful little “gems” which can be shared with one’s friends. This is the kind of beautiful stuff that makes mathematics truly interesting and exciting and though I have searched for a book like this for the last 35 years, this is by far the best in its class which I have found. Dr. John Aiken, Jan 5, 2003.
A professor in the University of Texas system has received a significant and lucrative honor:
Texas professor receives Norway’s Abel Prize for work in number theory
OSLO, Norway – An American professor at the University of Texas at Austin has won the 6 million kroner ($1 million) Abel Prize for mathematics.
The prize jury praised John Tate as “a prime architect” of number theory, a branch of mathematics that has played a key role in the development of modern computers.
The award citation issued Wednesday says Tate “has truly left a conspicuous imprint on modern mathematics” by advancing “one of (its) most elaborate and sophisticated branches.”
Tate’s scientific accomplishments span six decades. A wealth of essential mathematical ideas and constructions were initiated by Tate and later named after him, such as the Tate module, Tate curve, Tate cycle, Hodge-Tate decompositions, Tate cohomology, Serre-Tate parameter, Lubin-Tate group, Tate trace, Shafarevich-Tate group and Néron-Tate height.
In 2002-2003, Tate was a recipient of the Wolf Prize in Mathematics. The mathematician turned 85 this month and recently retired from his position as professor, becoming professor emeritus.
The annual Abel Prize was created by the Norwegian government in 2003 and is awarded to candidates who have contributed to the mathematical sciences. The winner is selected by an international committee of five mathematicians.
The prize will be given to Tate at a May 25 ceremony in Oslo.
This report includes information from The Associated Press and msnbc.com.
© 2010 msnbc.com
A good friend of the Metroplex Math Circle, Dr. Arthur Benjamin, has just released a new lecture course through the Teaching Company titled “Discrete Mathematics.” We have our pre-ordered copy and its seems to have the unique combination of humor and depth that we know from Dr. Benjamin’s excellent “mathemagic” presentations.
For any students just starting with Math Circles, they will benefit greatly from becoming familiar with the topics on these DVDs: number theory, combinatorics and graph theory.
Here is the description of the course from the Teaching Company:
Welcome to Discrete Mathematics, a subject that is off the beaten track that most of us followed in school but that has vital applications in computer science, cryptography, engineering, and problem solving of all types.
Most of the mathematics taught after elementary school is aimed at preparing students for one subject—calculus, which is the mathematics of how things grow and change continuously, like waves in the water or clouds in the sky. Discrete mathematics, on the other hand, deals with quantities that can be broken into neat little pieces, like pixels on a computer screen, the letters or numbers in a password, or directions on how to drive from one place to another.
While continuous mathematics resembles an old-fashioned analog clock, whose second hand sweeps continuously across a dial, discrete mathematics is like a digital watch, whose numbers proceed one second at a time. As a result, discrete mathematics achieves fascinating mathematical results using relatively simple means, such as counting.
Explore this modern realm of digital math in Discrete Mathematics, 24 mind-expanding lectures by veteran Teaching Company Professor Arthur T. Benjamin, an award-winning educator and mathemagician who has designed a course that is mathematically rigorous and yet entertaining and accessible to anyone with a basic knowledge of high school algebra.
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 , 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.
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.
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.
One of the only two numbers that need 9 cubes. (The other is 239.)
If negatives allowed,
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.
Sophie Germain prime: 2(23)+1 also prime.
The first pillar prime.
The second lecture of the season provided lessons in both applied mathematics and number theory. Dr. Bennette Harris began with an easy to follow introduction to binary and computer operations including XOR. He then defined the terms “encoding” and “encryption” and gave several examples of early encryption methods.
Dr. Harris described the difference between “strong” and “weak” encryption methods. This led to a deep dive into the popular RSA encryption method which depends on extremely large prime numbers. Dr. Harris used this as an opportunity to give the students a sense for mind-boggling large numbers and some of the very creative and efficient ways of determining whether or not a given number is a prime.
In this part of the discussion, Dr. Harris taught the students various techniques in modular arithmetic. Modular arithmetic is a tool that is not just useful in encryption but is critical for success in math contests and other pursuits.
Dr. Harris used several techniques to engage students including a series of 10 problems that he worked into the lecture for the students to solve. He also demonstrated the UBASIC program which very quickly found tremendously large prime numbers.
Dr. Harris has provided his slides from the lecture and answers to the problems. Members of the Metroplex Math Circle e-mail group can download these files from the group site. To join the e-mail group simply click below.
Here is the continuation of Dr. Bennett Harris’ problems to warm us up before his lecture on September 27th. Congratulations to Dominic for being the first to answer the previous set of problems in the comments. Another happy discovery is the fact that WordPress, which hosts this site, supports the use of code!
A solution for each of the following should either give the correct answer, or a technique for determining the answer in reasonable time with the assistance of a calculator.
6. Crack this secret message: “uifsfaxjmmacfabmnptuaopaqsppgtaupebz”
7. Code the alphabet a=0, b=1, …, z=25, space=26. We will encrypt each letter of the alphabet by mapping it to the letter 3 farther along in the alphabet. Thus, a maps to d, b maps to e, and so forth. The message “a bat is fat” would be encrypted as “dcedwclvciodw” in this scheme. Find a mathematical function f(x) such that y=f(x) gives the correct encryption for every value x = 0,1,2,…,26. Test your function by encrypting “the toy boat floats”.
8. The purpose of these two questions is to provide points of comparison for the questions that follow:
How many particles are there in the known universe?
How many microseconds have elapsed since the beginning of the universe?
9. How long (approximately will do) would it take to calculate at 1,000,000 calculations per second? How many digits does the answer have? If p is 200 digits long, and if q is 200 digits long, about how many digits are there in pq?
10. The number – 1 is known to be prime. Estimate how long it would take a computer to demonstrate this by repeated division, at 1,000,000 divisions per second.