2. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Surprises in Experimental Mathematics Michael I. Shamos School of Computer Science Carnegie Mellon University

3. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Mathematical Discovery Where do theorems come from? Theorem easy to conjecture, proof is hard –Fermat’s last theorem, four-color theorem Theorem surprising, but results from a logical investigation –Fundamental theorem of algebra Theorem difficult to invent, straightforward to prove –Hadamard three-circles theorem –Where do these come from?

4. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Hadamard Three-Circles Theorem If f(z) is holomorphic (complex differentiable) on the annulus centered at the origin and then a b r How was this theorem ever conjectured?

5. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Outline The problem –Closed-form expression for The approach –Build a catalog of real-valued expressions indexed by first 20 digits –Equivalent expressions will “collide” –Look up 1.20205690315959428539 The discoveries –The Partial Sum Theorem –Overcounting functions –How many ways can n be expressed as an integer power k j ? –Expression for –...

6. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS The Problem Closed form expressions for values of the zeta function Euler found an expression for all even values of s: No expression is known for even a single odd value, e.g. is a rational multiple of

7. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS The Catalog Some values of :

8. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Other Catalogs Sloane’s Encyclopedia of Integer SequencesEncyclopedia –Terrific, but for integer sequences, not reals Plouffe’s InverterInverter –Huge (215 million entries), but not “natural” expressions from actual mathematical work Simon Fraser Inverse Symbolic CalculatorInverse Symbolic Calculator –50 million constants

9. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Discovery A where is the number of primes  k Is this a coincidence? Why the factor of 2? Is there a general principle at work? –In fact,

10. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Observation is a partial sum function, i.e., So can be rewritten as: where f and g are “related” More generally, is the partial sum function of the indicator function of the property “primeness”:

11. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS The Partial Sum Theorem Given a sequence S of complex numbers s(k), let be the sequence of partial sums of S. Given a function f, if certain convergence criteria are satisfied, then where (the partial tails of f) is a transform of f independent of s & t PARTIAL SUMS OF s( j ) PARTIAL TAILS OF f ( k ) (New)

12. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Partial Sum Functions Many sequences are partial sum functions: the harmonic function generalized harmonic function Actually, every sequence is the partial sum function of some other sequence

13. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Some Partial Sum Transforms

14. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Some Partial Sum Transforms

15. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Partial Sum Theorem (Proof) Consider the upper triangular matrix HEADS OF s = t(i) TAILS OF f = g(j) Column sums are Row sums are

16. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS The Convergence Criteria 1.All sums g(k) converge 2. converges; and 3. iff Proof: By Markoff’s theorem on convergence of double series

17. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Further Applications The number of perfect n th powers  k is The number of positive integers powers of a  k is Therefore, by inspection, * (Old) * * * *

18. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS The Inverse Transform Given g(j), how can we find f(k)? Since g is a sum of f’s, f is the sequence of finite differences of g : Subtracting,

19. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Some Inverse Transforms

20. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Some Inverse Transforms

21. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS The Partial Integral Theorem Given a function s(x), let t(y) be the “left integral” of s : Given a function f(y), if certain convergence criteria are satisfied, then where (the right integral of f) is a transform of f independent of s & t LEFT INTEGRAL OF s( x ) RIGHT INTEGRAL OF f ( y ) *

22. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Example Therefore, Note: if s(x) is a probability density then t(y) is its cumulative distribution function

23. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Example Consider the special case in which This implies. So

24. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Example Bessel functionStruve function What about Risch’s theorem? Risch, R. “The Solution of the Problem of Integration in Finite Terms.” Bull. Amer. Math. Soc., 1-76, 605-608, 1970. Mathematica gives up. Mathematica has no problem

25. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Discovery B Is there a general principle at work? must exceed, but by how much? –In fact,

26. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Overcounting Functions 1.Every term of S + occurs at least once in S. 2.In general, S + overcounts S, since some terms of S occur many times in S + 3.If K g (k) is the number of times f(k) is included in S +, then where K g (k) depends only on g and not on f. where ranges over the natural numbers and Let,

27. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Examples Let g(k, j) = k + j. How many ordered pairs (k, j) of natural numbers give k + j = n? Answer: K k+j (n) = n - 1 Therefore, by inspection,

28. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Examples Let g(k, j) = k j. How many ordered pairs (k, j) give k j = n? Answer: K kj (n) = d(n), the number of divisors of n. Therefore, by inspection

29. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Enumerating Non-Trivial Powers Let g(k, j) = k j. How many ordered pairs (k, j) give k j = n? Or, how many ways K(n) can n be expressed as a positive integral power of a positive integer? Let be the prime factorization of n n can be a non-trivial power of an integer > 1 iff G(n) = gcd(e 1, e 2,... ) exceeds 1; otherwise K(n) = 1. Suppose b > 1 divides G(n). Then, where each of the e i /b is a natural number, so n is the b th power of a natural number Suppose c > 1 does not divide G(n). Then at least one of the exponents e i /c is not a natural number and n is not the c th power of a natural number. Therefore, *

30. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS A Remarkable Series Let. Then yields the “overcounting” function (Old) *

31. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Goldbach’s Theorem In 1729, Christian Goldbach proved that 4242 4343 4... 8282 16 2 8383 16 3

32. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Discovery C (Old) What is * *

33. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Discovery D Since the partial sum function of is, where is the number of distinct prime factors of k *

34. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Discovery E For c > 1 real and p prime, In particular,

35. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Results The counting function K max (n) of max(k, j) is 2n-1. So The counting function K lcm (n) of lcm(k, j) is d(n 2 ). So

36. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS The First-Digit Phenomenon Given a random integer, what is the probability that its leading digit is a 1? Answer: depends on the distribution from which k is chosen If k is chosen uniformly in [1, n], then let p(d, n) be the probability that the leading digit of k is d For n = 19 +, 5/9 < p(1,n) <.579; 1/19  p(9,n) < 1/18 For n = 9 +, p(1,n) = 1/9; p(9,n) = 1/9 The “average” is log 10 (1+1/d) {.301,.176,.124,.097,.079,.066,.058,.051,.046}

37. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Relative Digit Frequency (Benford’s Law) log 10 (1+1/d)

38. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Relative Digit Frequency

39. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS First-Digit Phenomenon

40. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Major Ideas For mathematicians: –How to populate the catalog –How to generalize from discoveries For computer scientists: –Use in symbolic manipulation systems For data miners: –How to mine the catalog, i.e. how to find new relations For statisticians: –How to use the fact that where P is the cumulative distribution of density p

41. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS A Parting Philosophy “The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there was never any other object for it.” Jacques Hadamard (as quoted by Borel in 1928)

42. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Q A &

43. Results n! is the nearest integer to

44. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Hypergometric Functions The are solutions of the hypergeometric differential equation: where

45. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Partial Sum Theorem (Proof) Consider the upper triangular matrix The sum of row i is The sum of column j is The sum of the row sums equals the sum of the columns sums precisely when the conditions of Markoff’s theorem are satisfied. QED

46. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Correspondence with Plouffe

47. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS First-Digit Phenomenon SOURCE: SIMON PLOUFFESIMON PLOUFFE

48. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS First Digit Frequency

49. MCGILL UNIVERSITY SEPTEMBER 21, 2007 COPYRIGHT © 2007 MICHAEL I. SHAMOS Correspondence with Plouffe