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Taxicabs and Sums of Two Cubes

Taxicabs and Sums of Two Cubes. An Excursion in Number Theory. Undergraduate Lecture in Number Theory Hunter College of CUNY Tuesday, March 12, 2013. Joseph H. Silverman, Brown University. kingdom across the Atlantic,. mathematician named Hardy. Our Story Begins….

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Taxicabs and Sums of Two Cubes

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  1. Taxicabs and Sums of Two Cubes An Excursion in Number Theory Undergraduate Lecture in Number Theory Hunter College of CUNY Tuesday, March 12, 2013 Joseph H. Silverman, Brown University

  2. kingdom across the Atlantic, mathematician named Hardy Our Story Begins… A long time ago in a galaxy far, far away, a Jedi knight named Luke received a mysterious package. This package, from a young Indian office clerk namedRamanujan,contained pages of scribbled mathematical formulas. Some of the formulas were well-know exercises. Others looked preposterous or wildly implausible. But Hardy and a colleague managed to prove some of these amazing formulas and they realized that Ramanujan was a mathematical genius of the first order.

  3. Our Story Continues… Hardy arranged for Ramanujan to come to England. Ramanujan arrived in 1914 and over the next six years he produced a corpus of brilliant mathematical work in number theory, combinatorics, and other areas. In 1918, at the age of 30, he was elected a Fellow of the Royal Society, one of the youngest to ever be elected. Unfortunately, in the cold, damp climate of England, Ramanujan contracted tuberculosis. He returned to India in 1920 and died shortly thereafter.

  4. 1729 A “Dull” Taxicab Number Throughout his life, Ramanujan considered numbers to be his personal friends. One day when Ramanujan was in the hospital, Hardy arrived for a visit and remarked: The number of my taxicab was 1729. It seemed to me rather a dull number. To which Ramanujan replied: No, Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.

  5. equals13 + 123 equals93 + 103 An Interesting Taxicab Number 1729 1729is a sum of two cubes in two different ways

  6. Sums of Two Cubes The taxicab number 1729 is a sum of two cubes in two different ways. Can we find a number that is a sum of two cubes in three different ways? [When counting solutions, we treat a3+b3 and b3+a3 as the same.] The answer is yes: 4104 = 163 + 23 = 153 + 93 = (–12)3 + 183. Of course, Ramanujan really meant us to use only positive integers: 87,539,319 = 4363 + 1673 = 4233 + 2283 = 4143 + 2553. How about four different ways? And five different ways? And six different ways? And seven different ways? …

  7. Sums of Two Cubes in Lots of Ways Motivating Question Are there numbers that can be written as a sum of two (positive) cubes in lots of different ways? The answer, as we shall see, involves a fascinating blend of geometry, algebra and number theory. And at the risk of prematurely revealing the punchline, the answer to our question is YES , sort of , well, actually MAYBE YES, MAYBE NO.

  8. Taxicab Equations and Taxicab Curves Motivating Question as an Equation Are there numbers A so that the taxicabequation X3 + Y3 = A has lots of solutions (x,y) using (positive) integers x and y? Switching from algebra to geometry, the equation X3 + Y3 = A describes a “taxicab curve” in the XY-plane.

  9. The Geometry of a Taxicab Curve So let’s start with an easier question. What are the solutions to the equation X3 + Y3 = A in real numbers? In other words, what does the graph of X3 + Y3 = A look like?

  10. The Taxicab Curve: X3 + Y3 = A

  11. X = Y P Q R L “Adding” Points on a Taxicab Curve P + Q

  12. X = Y P R “Doubling” a Point on a Taxicab Curve P+P = 2P Tangent line to C at the point P

  13. X = Y P Q Where is the “Missing” Point? P = (a,b) and Q = (b,a) L There is no third intersection point!!! What to do, what to do, what to ................................................?

  14. X = Y P Q O A Pesky Extra Point “at Infinity” Syllogism: 1st Premise: We want a third intersection point. 2nd Premise: Mathematicians always get what they want. Conclusion: Since there is no actual third point, we’ll simply pretend that there is a third point hiding out “at infinity”. We’ll call that third point O. L O is an extra point “at infinity”It lies on the curve C and the line L.

  15. X = Y P = (x1,y1) Q = (x2,y2) R L A Messy Formula for Adding Points P + Q Using a little bit of geometry and a little bit of algebra, we can find a formula for the sum of P and Q. It is a messy formula, but quite practical for computations.

  16. We can add and subtract them: We can double them and triple them and on and on and on… An Example: The Taxicab CurveX3 + Y3 = 1729 Start with Ramanujan’s two points: P = (1,12) and Q = (9,10)

  17. Easy to prove Surprisingly difficult Is “Addition” Really Addition? Adding points on the taxicab curves is certainly very different from ordinary addition of numbers. But “taxicab addition” and ordinary addition do share many properties. Let O denote the extra point “at infinity” and for any point P = (x,y) on the curve, let –P be the reflected point (y,x). Properties of Taxicab Addition: P + O = O + P = P P + (–P) = O P + Q = Q + P (P + Q) + R = P + (Q + R) identity element inverse commutative law associative law In mathematical terminology, the points on the taxicab curve form aGROUP.

  18. Elliptic Curves Curves with an addition law are called Elliptic Curves • Elliptic curves and functions on elliptic curves play an important role in many branches of mathematics and other sciences, including: • Number Theory • Algebraic Geometry • Cryptography • Topology • Physics

  19. This is obvious from the formula for P+Q: Adding Rational Points Gives More Rational Points Taxicab addition has one other very important property: If the coordinates of P = (x1,y1) and Q = (x2,y2) are rational numbers, then the coordinates of P+Q are also rational numbers. The formula is messy, but if A is an integer and if x1, y1, x2, y2 are rational numbers, then the coordinates of P+Q are clearly rational numbers.

  20. The Group of Rational Points This means that we can add and subtract points in the set C(Q) = { (x,y)  C : x and y are rational numbers }  {O} and stay within this set. Thus C(Q) is also a group. One of the fundamental theorems of the 20th century says that we can get every point in C(Q) by repeated addition and subtraction using a finite starting set. Mordell’s Theorem (1922): There is a finite set of points { P1, P2, …, Pr } in C(Q) so that every point in C(Q) can be found by repeatedly adding and subtracting P1, P2, …, Pr. In other words, for every point P in C(Q), we can find integers n1,n2,…, nr so that P = n1P1 + n2P2 + … + nrPr.

  21. And every rational point on the taxicab curve X3 + Y3 = 1729 can be obtained by using the two generating points P = (1,12) and Q = (9,10). Examples of Groups of Rational Points For example, every rational point on the curve X3 + Y3 = 7 is equal to some multiple of the single generating point (2, –1). So we now understand how to find lots of solutions to X3 + Y3 = A using rational numbers x and y, but our original problem was to find lots of solutions using integers.

  22. Use P to find some points with rational coordinates: Turning Rational Numbers Into Integers How can we change rational numbers into integers? Answer: Multiply by a common denominator. Start with the point P = (2, –1) on the curve X3 + Y3 = 7. Now multiply everything by 33.383 to clear the denominators!

  23. A Taxicab Curve With Three Integer Points Multiply by 33.383 to clear the denominators! We have constructed a taxicab number A = 7.33.383 = 10,370,808 that is a sum of two cubes in three different ways: 2283 + (–114)3=1903 + 1523=(– 51)3 + 2193=10,370,808

  24. We simply start with four points on the curve X3 + Y3 = 7, and clear their denominators to get a taxicab number A = 7. 33. 383. 1833 =63,557,362,007,496 that is a sum of two cubes in four different ways. And so on. If we start with N points P, 2P, 3P, 4P, …, NP and clear all of their denominators, then we will get a (very large) taxicab number that is a sum of two cubes in N different ways. Taxicab Curves With Lots of Integer Points Suppose that we want a taxicab curve with four integer points.

  25. But Ramanujan Used Positive Integers… That’s okay, because it is possible to prove that in the list of points P, 2P, 3P, 4P, 5P, 6P, 7P, … there are infinitely many of them whose x and y coordinates are both positive. So we can take N of these “positive” points from the list and clear all their denominators. This provides an affirmative answer to our original question. Pick any number N. Then we can find a taxicab number A so that the taxicab equation X3 + Y3 = A has at least N different solutions (x,y) using positive integers x and y.

  26. Finding the Smallest Taxicab Numbers The N’th Taxicab Number is the smallest number A so that we can write A as a sum of two positive cubes in at least N different ways. It is not easy to determine Taxi(N) because the numbers get very large, so it is hard to check that there are no smaller ones. Here is the current list. Taxi(1) = 2Taxi(2) = 1729Taxi(3) = 87539319Taxi(4) = 6963472309248Taxi(5) = 48988659276962496Taxi(6) = 24153319581254312065344 Discovered in:16571957199119972008 Maybe you can find the next one!

  27. Are We Really Done? What we have done is take a lot of solutions using rational numbers and cleared their denominators. This answers the original question, but… it feels as if we’ve cheated. Suppose that we want to find taxicab numbers that are truly integral and that do not come from clearing denominators. How can we tell if we’ve cheated? Well, if A comes from clearing denominators, then the x and y values will have a large common factor. New Version of the Motivating Question Are there taxicab numbers A for which the equation X3 + Y3 = A has lots of solutions (x,y) using positive integers so that x and y have no common factor?

  28. Taxicab Solutions With No Common Factor Is there a taxicab number A withtwo positive no-common-factor solutions? Yes, Ramanujan gave us one: 1729 = 13 + 123 = 93 + 103. Is there a taxicab number A withthree positive no-common-factor solutions? Yes, Paul Vojta found one in 1983. At the time he was a graduate student and he discovered this taxicab number using an early desktop IBM PC! 15,170,835,645 equals5173 + 24683 = 7093 + 24563 = 17333 + 21523

  29. Taxicab Solutions With No Common Factor How about a taxicab number A withfour positive no-common-factor solutions? Yes, there’s one of those, too, discovered (independently) by Stuart Gascoigne and Duncan Moore just 10 years ago. 1,801,049,058,342,701,083 equals 922273 + 12165003and 1366353 + 12161023and 3419953 + 12076023and 6002593 + 11658843

  30. A Taxicab Challenge Find a taxicab number A with five positive no-common-factor solutions. Taxicab Solutions With No Common Factor Is there a taxicab number A withfive positive no-common-factor solutions? NO ONE KNOWS!!!!! * * Or prove that none exist!!!

  31. Futurama Epilogue Bender is a Bending-Unit: Chassis # 1729 Serial # 2716057 Bender's serial number 2716057 is, of course, a sum of two cubes: 2716057 = 952³ + (-951)³. So take Bender’s advice: “Sums of Cubes are everywhere. Don’t leave home without one!”

  32. Taxicabs and Sums of Two Cubes Joseph H. Silverman, Brown University

  33. Taxicabs and Sums of Two Cubes Joseph H. Silverman, Brown University

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