THE UNREASONABLE USEFULNESS OF PRIME NUMBERS

THE UNREASONABLE USEFULNESS OF PRIME NUMBERS April 11, 2006

The teaching of Jack McLaughlin Jack McLaughlin was the best teacher I had in my undergraduate and graduate student career. … I never thought of mathematics as `beautiful’ until I saw Jack's lectures on linear algebra … In my mathematical research, teaching, and administrative career, which has been spent at four major universities, I have never encountered a better teacher of undergraduate mathematics than Jack McLaughlin.

The problems … he made the greatest impression of all my teachers. … The challenge came in the form of the problems. I loved them. I have kept them … and assigned them to my own students. … I remember with delight one of my undergraduates who initially looked undistinguished, but blossomed under the effect of the McLaughlin problems. (He is now a research mathematician.)

a life well spent It is surely true that McLaughlin was … an extraordinary teacher. His lectures were, it seemed, perfectly conceived and flawlessly presented. … the demands he made on the honors students were extraordinary. In particular, the problem assignments were enormously challenging. … the problems took up most of one's life, but it was a life well spent since the assignments, like the lectures, were masterpieces.

Handwriting improvement Despite my incredibly sloppy handwriting and note-taking, my notebooks from his courses present the material with textbook clarity. Apparently I improved my handwriting just for his courses. … After 21 years, few of my professors at Michigan are as memorable as Prof. McLaughlin. There should be no doubt in anybody's mind that he is remembered and appreciated by his students indefinitely.

Bach at the organ Professor McLaughlin at the board is like Bach at the organ. … Fifteen years later I still have my class notes … Professor McLaughlin had a profound effect on my career. … After a while I didn't care about the grade. The whole point was to learn - failing to solve a problem was part of the process. Who could assign graduate level problems to sophomores and have them look forward to it?

Demanding and motivating He was probably the best teacher I ever had; his courses were the most demanding I ever had; and under his tutelage I learned what it was to be a mathematician. **************** … his technique was very motivating … I have kept all my notebooks and homework problems from Dr. McLaughlin's courses. His were the only courses whose materials I kept …

samizdat … the most extraordinary thing about his teaching, undergraduate or graduate … was his ability to deliver lectures of astonishing clarity and insight day after day after day. … his lectures were seamless narratives, raising questions, examining them from every angle, considering examples, and then resolving the questions. … There can't be many like Jack, whose lecture notes circulate in photo-copy among mathematicians like a kind of samizdat.

Preparation

What Jack McLaughlin taught Mathematics is beautiful Butting your head into the wall is rewarding (it should be the right wall though)

How does one mentor new researchers? Strategies: Give support and be a coach Be a stern taskmaster Let ‘em alone (they will find their own way)

Decisive advice Maybe, or maybe not!

Know the student Be flexible Be available Work at it

What’s going on?! • 1/3 of women interviewed who had exited science cited a lack of guidance as the major factor leading to the exit decision • None of the men interviewed identified lack of mentoring as a factor influencing exit • 73% of all the women interviewed described situations where either positive mentors advanced their careers or the indifference, and even hostility, of potential mentors impeded their careers Based on interviews with 102 men and women with S&E degrees of whom half remained employed. Leaving Science: Occupational Exit of Scientists and Engineers,Anne E. Preston, Haverford College, June 2003

Think about it! Both thought and a new perspective are needed.

Prime numbers A whole number at least 2 is prime if it is not the product of two smaller whole numbers. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, … are prime. 2 is the only even prime. 4 = 2 x 2, 6 = 2 x 3, 9 = 3 x 3, are not. There are infinitely many. Euclid knew a proof:

There’s always another prime One can always get one more prime: multiply the ones you have together and add one. The smallest factor (other than 1) of the number obtained will be prime. 2 x 3 x 5 x 7 = 210. 210 + 1 = 211 is prime. 2 x 3 x 5 x 7 x 211 + 1 = 44311 = 73 x 607 73 is a “new” prime.

The new and the unknown There are arbitrarily long arithmetic progressions consisting of primes: 5, 11, 17, 23, 29 is such a progression (each time one adds 6). Not known: are there infinitely many pairs like 11, 13; 17,19; 29, 31; 41, 43 consisting of consecutive odd integers that are both prime. These pairs are called “twin primes.” This is strongly conjectured.

Baseball Paul Erdos was a very prolific mathematician with many co-authors who was intrigued by prime numbers. His co-authors are said to have Erdos number 1. Co-authors of those with number 1 are said to have Erdos number 2, and co-authors of those with number 2 are said to have Erdos number 3, and so forth. There is a sense in which the baseball player Hank Aaron has Erdos number 1.

Ruth-Aaron pairs When Aaron broke Babe Ruth’s record for number of home runs (714), number theorist Carl Pomerance noticed that 714 = 2 x 3 x 7 x 17, and 2 + 3 + 7 + 17 = 29. But 715 = 5 x 11 x 13, and 5 + 11 + 13 = 29. These consecutive numbers have the same sum for their prime factors. Pomerance named these “Ruth-Aaron” pairs and made a conjecture about their density. Erdos proved it! Erdos and Hank Aaron received honorary degrees from the University of Georgia.

Number 1 They jointly autographed a baseball. Arguably, this gives Hank Aaron Erdos number 1.

Arithmetic mod 12 In telling time, one ignores multiples of 12. If it is 4 o’clock now, in 27 hours it will be 4 + 27 - 12 - 12 = 7 o’clock. Multiples of 12 (hours) don’t matter. 12 = 0, in the sense that adding 12 hours keeps the time the same! Working mod 12 we ignore multiples of 12.

Arithmetic mod 360 In working with angles in degrees, multiples of 360 can be ignored. If you rotate a compass needle first 190º and then 210º, the effect is the same as rotating it 190 + 210 - 360 = 40º. In this context, 360 = 0 !

Arithmetic mod 10 What happens when you add or multiply two numbers and you only know the last digits? …?7 …?7 + …?9 x …?9 ________ _______

Arithmetic mod 10 What happens when you add or multiply two numbers and you only know the last digits? …?7 …?7 + …?9 x …?9 ________ _______ …?6 …?3 You still know the last digit of the answer!

Arithmetic the easy way You can do arithmetic mod 10. It turns out to make sense. You add and multiply the numbers 0 through 9 as usual, and only keep track of the last digit. 5 + 9 = 4 4 x 8 = 2 7 + 6 = 3 7 x 3 = 1 9 + 1 = 0 2 x 5 = 0 In this world, 10 = 0 !

Choose your ignorance Arithmetic makes sense not only mod 10, but mod 12, mod 360 or mod any number you choose. You can decide what number you want to “ignore.” The number you choose becomes 0 in your arithmetic. In mod 2 arithmetic one has 0 + 0 = 0 0 x 0 = 0 0 + 1 = 1 0 x 1 = 0 1 + 0 = 1 1 x 0 = 0 1 + 1 = 0 1 x 1 = 1

Rings In working with numbers mod 12 it is natural to picture them arranged in a circle like the numerals on a clock (the old-fashioned kind). A set of elements for which one has “good” notions of addition and multiplication is called a “ring,” and working with the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 mod 12 gives an example.

Ring 0 11 1 10 2 9 3 8 4 7 5 6

More rings The integers, the real numbers, and the complex numbers also give examples of rings. The ones we’ll consider contain an element 1 such that 1 x r = r x 1 = r for all r in the ring. There are many other examples, including A huge family of rings in which some prime p = 1 + 1 + … + 1 (p copies) is 0.

The Chinese remainder theorem Knowing a number mod 30 (the remainder when it is divided by 30) is the same as knowing the number mod 2, 3, and 5. Example: The remainder mod 2 is 1. The remainder mod 3 is 2. The remainder mod 5 is 3. What is the remainder mod 30?

The Chinese remainder theorem Knowing a number mod 30 (the remainder when it is divided By 30) is the same as knowing the number mod 2, 3, and 5. Example: The remainder mod 2 is 1. The remainder mod 3 is 2. The remainder mod 5 is 3. What is the remainder mod 30? Answer:23

… with more primes An entirely similar statement holds for any finite set of prime numbers. For example, for 2, 3, 5, 7, 11 one has that there is one and only one whole number less than 2 x 3 x 5 x 7 x 11 = 2310 with specified remainders mod 2, 3, 5, 7, and 11.

Parallel computing Mathematical computing problems can be viewed in terms of manipulating whole numbers to produce answers that are whole numbers. Suppose one knows that the answer to a problem involves at most 50 digits. Then one can get the answer by solving the problem mod many different primes: enough primes so that their product has more than 50 digits.

Small pieces If one takes the first 32 prime numbers, from 2 to 131, and multiplies them together one gets 525896479052627740771371797072411912900610967452630 which has 51 digits. The computing problem can therefore be solved on 32 CPUs simultaneously, each working mod a different one of these 32 primes. Prime numbers give a way of breaking any problem into small pieces that can be solved separately.

Codes Make the message into a number, perhaps by a substitution like A — 101 W — 123 B — 102 … X — 124 C — 103 Y — 125 D — 104 Z — 126 Think of the message to be sent as a large number, perhaps a 400 digit number.

Transforming in mysterious ways What is needed is a way of manipulating 400 digit numbers, transforming them into new numbers: these are the coded messages. Primes can be used as follows: one first finds two 200 digit primes, p and q, and multiplies them together. Call the result N. Think of possible messages as remainders mod N. One also finds numbers E and D such that ED = 1 mod (p-1)(q-1).

The power of encoding One encodes the message M, a large number, by calculating M to the E th power mod N. The answer is the coded message C. One decodes C by calculating C to the D th power mod N. Someone who knows N and E still cannot decode, unless told the value of D — unless that person can figure out the factorization N = pq.

An example with small primes Let p = 57689 and q = 71861. Then N = pq = 4145589229, and s = (p-1)(q-1) = 4145459580. One may choose E = 2174596739, and then D = 1780436939. To encode the message M = 2347689453 one raises M to the E th power mod N. The result is 1874445480. To decode, one raises 1874445480 to the 1780436939 th power mod N. The number put in and the number that one gets by this method seem completely unrelated.

Is a code of this type safe? Breaking these codes is related to factoring very large numbers. In consequence, there has been a great interest in this problem. Factoring 400 digit numbers into two 200 digit primes seems out of reach to all current methods, even given billions of years for the attempt.

Fermat’s little theorem The theory behind these codes depends on a fact known as Fermat’s little theorem. It says that if p is prime, and one raises any nonzero integer smaller than p to the (p-1)st power, the result is one. For example, if p = 13, one has that the twelfth power of 6 mod 13 is 1. 6 can be replaced in this statement by any integer from 1 to 12.

Go for the ring Part of my research deals with understanding solutions of large systems of polynomial equations over what might be the complex numbers — or something else. One can think about solutions in many ways. One is to study the graph, called an algebraic set. It has geometry. It has a dimension. Another is to “force” the equations to hold in a specially constructed ring, and then study the ring.

Graphs Here are graphs of some quadric equations In real 3 space: With simulltaneous equations, graphs are intersected.

Planes forced to meet in a line Two planes in 3-space are forced to meet in a line. Add the dimensions of The objects that are intersecting and subtract the dimension of the ambient space. 2 + 2 - 3 = 1

This holds a lot more generally The same fact about dimensions of intersections holds for any two algebraic sets over the complex numbers that meet at a point P (one uses their dimensions near P) when the sets are in complex n-space. Linearity is not needed! When the ambient space is not so nice (singular), one can sometimes get results like these with the right additional conditions: such results tend to be difficult.

Work mod p for many values of p Even starting with complex numbers, when working with rings one can make a transition to studying infinitely many other rings, in each of which a different prime integer has been forced to be zero. Each of these rings contains a ring that is just like the integers mod p for some prime p. Some very difficult results controlling dimension can be proved in this way, and no other proof is known.

Force p = 0 for an easy life For many problems, knowing that p = 0 makes life easier. One of the reasons for this is that when p = 0, one has the formula The terms that involve both variables are not present! One gets a contradiction out of equations by raising them to high powers in which the exponent is a power of p.

Ubiquity Many problems that at first glance do not seem to be of this type can be transformed into one of this kind.

Let f, g, and h be three polynomials in two variables with complex (or rational) coefficients. Then there are always polynomials A, B, and C such that The easiest proof by far is by transition to rings where some prime p = 0.

THE UNREASONABLE USEFULNESS OF PRIME NUMBERS