Surprises in Experimental Mathematics

1. Surprises inExperimental Mathematics Michael I. Shamos School of Computer Science Carnegie Mellon University

2. 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?

3. • • • Hadamard Three-Circles Theorem • If f(z) is holomorphic (complex differentiable) on the annulus centered at the origin and • then b How was this theorem ever conjectured? r a

4. 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 kj ? • Expression for • . . .

5. is a rational multiple of 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.

6. The Catalog • Some values of :

7. Other Catalogs • Sloane’s Encyclopedia of Integer Sequences • Terrific, but for integer sequences, not reals • Plouffe’s Inverter • Huge (215 million entries), but not “natural” expressions from actual mathematical work • Simon Fraser Inverse Symbolic Calculator • 50 million constants

8. whereis the number of primes £ k • In fact, Discovery A • Is this a coincidence? • Why the factor of 2? • Is there a general principle at work?

9. More generally, is the partial sum function of the indicator function of the property “primeness”: So can be rewritten as: where f and g are “related” Observation is a partial sum function, i.e.,

10. PARTIAL TAILS OF f(k) PARTIAL SUMS OF s(j) The Partial Sum Theorem (New) • 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, thenwhere(the partial tails of f) is a transform of f independent of s & t

11. 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

12. Some Partial Sum Transforms

13. Some Partial Sum Transforms

14. TAILS OF f = g(j) Row sums are HEADS OF s = t(i) Column sums are Partial Sum Theorem (Proof) • Consider the upper triangular matrix

15. The Convergence Criteria • All sums g(k) converge • converges; and iff Proof: By Markoff’s theorem on convergence of double series

16. Further Applications • The number of perfect nth powers  k is • The number of positive integers powers of a k is • Therefore, by inspection, * * * * * (Old)

17. 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,

18. Some Inverse Transforms

19. Some Inverse Transforms

20. RIGHT INTEGRAL OF f(y) LEFT INTEGRAL OF s(x) 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, thenwhere(the right integral of f) is a transform of f independent of s & t

21. Example Therefore, Note: if s(x) is a probability density then t(y) is its cumulative distribution function

22. Consider the special case in which This implies . So Example

23. Struve function Bessel function Example Mathematica gives up. Mathematica has no problem 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.

24. In fact, Discovery B • Is there a general principle at work? must exceed , but by how much?

25. and Let , where ranges over the natural numbers Overcounting Functions • Every term of S+ occurs at least once in S. • In general, S+ overcounts S, since some terms of S occur many times in S+ • If Kg(k) is the number of times f(k) is included in S+, thenwhere Kg(k) depends only on g and not on f .

26. Examples • Let g(k, j) = k + j . How many ordered pairs (k, j) of natural numbers give k + j = n? Answer: Kk+j (n) = n - 1 • Therefore, by inspection,

27. Examples • Let g(k, j) = k• j . How many ordered pairs (k, j) give k • j = n? Answer: Kk•j (n) =d(n), the number of divisors of n . • Therefore, by inspection

28. Enumerating Non-Trivial Powers • Let g(k, j) = kj. How many ordered pairs (k, j) give kj = 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 iffG(n) = gcd(e1, e2, . . . ) exceeds 1; otherwise K(n) = 1. • Suppose b >1 divides G(n). Then ,where each of the ei /b is a natural number, so n is the bth power of a natural number • Suppose c > 1 does not divide G(n). Then at least one of the exponents ei /c is not a natural number and n is not the cth power of a natural number. Therefore, *

29. A Remarkable Series (Old) • Let . Then the “overcounting” function yields *

30. 42 43 . . . 44 82 83 163 162 Goldbach’s Theorem • In 1729, Christian Goldbach proved that

31. * * Discovery C (Old) What is

32. where is the number of distinct prime factors of k Discovery D * , • Since the partial sum function of is

33. Discovery E For c > 1 real and p prime, In particular,

34. Results The counting function Kmax(n) of max(k, j) is 2n-1. So The counting function Klcm(n) of lcm(k, j) is d(n2). So

35. 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 log10(1+1/d) • {.301, .176, .124, .097, .079, .066, .058, .051, .046}

36. Relative Digit Frequency (Benford’s Law) log10(1+1/d)

37. Relative Digit Frequency

38. First-Digit Phenomenon

39. 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 thatwhere P is the cumulative distribution of density p

40. 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)

41. Q A &

42. Results n! is the nearest integer to

43. where Hypergometric Functions • The are solutions of the hypergeometric differential equation:

44. 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

45. Correspondence with Plouffe

46. First-Digit Phenomenon SOURCE: SIMON PLOUFFE

47. First Digit Frequency

48. Correspondence with Plouffe