1. Modular Juggling with Fermat Stephen Harnish Professor of Mathematics Bluffton University harnishs@bluffton.edu Miami University 36 th Annual Mathematics & Statistics Conference: Recreational Mathematics September 26-27, 2008 Archive of Bluffton math seminar documents: http://www.bluffton.edu/mcst/dept/seminar_docs/

2. Modular Juggling with Fermat

3. Theorem 1: (Euler) The sequence has no equal initial and middle sums. Theorem 2: (Dirichlet) The sequence has no equal initial and middle sums. Classical Results

4. Initial and Middle Sums of Sequences Note that sequence {1, 2, 3, 4, …} has numerous initial sums that equal middle sums: (1 + 2) = 3 = (3) (1 + 2 + 3 + 4 + 5) = 15 = (7 + 8) (1 + 2 + 3 + 4 + 5 + 6) = 21 = (10 + 11) (1 + 2 + 3 + 4 + 5 + 6) = 21 = (6 + 7 + 8)

5. Sequence Sums Definition: For the sequence an initial sum is any value of the form for some integer k and a middle sum is any value of the form for some integers j and k, where the length of a middle sum is.

6. Initial and Middle Sums of Sequences Note that sequence {1, 2, 3, 4, …} has numerous initial sums that equal middle sums: (1 + 2) = 3 = (3) (1 + 2 + 3 + 4 + 5) = 15 = (7 + 8) (1 + 2 + 3 + 4 + 5 + 6) = 21 = (10 + 11) (1 + 2 + 3 + 4 + 5 + 6) = 21 = (6 + 7 + 8)

7. Initial and Middle Sums of Sequences--Fibonacci Note that sequence {1, 1, 2, 3, 5, 8, 13…} has the following initial sums: (1) = 1 = (1) (1 + 1) = 2 = (2) (1 + 1 + 2) = 4 (1 + 1 + 2 + 3) = 7 (1 + 1 + 2 + 3 + 5) = 12 (1 + 1 + 2 + 3 + 5 + 8) = 20

8. Juggling History 1994 to 1781 (BCE)—first depiction on the 15 th Beni Hassan tomb of an unknown prince from Middle Kingdom Egypt. The Science of Juggling 1903—psychology and learning rates 1940’s—computers predict trajectories 1970’s—Claude Shannon’s juggling machines at MIT The Math of Juggling 1985—Increased mathematical analysis via site-swap notation (independently developed by Klimek, Tiemann, and Day) For Further Reference: Buhler, Eisenbud, Graham & Wright’s “Juggling Drops and Descents” in The Am. Math. Monthly, June-July 1994. Beek and Lewbel’s “The Science of Juggling” Scientific American, Nov. 95. Burkard Polster’s The Mathematics of Juggling, Springer, 2003. Juggling Lab at http://jugglinglab.sourceforge.net/http://jugglinglab.sourceforge.net/

9. Juggling Patterns (via Juggling Lab)

10. Thirteen-ball Cascade

11. A 30-ball pattern of period-15 named: “uuuuuuuuuzwwsqr” using standard site-swap notation

13. 531

14. Several period-5, 2-ball patterns 90001122233052014113

15. A Story Relating Juggling with Number Theory

16. A Tale of Two Kingdoms First Studied by E. Tamref Values of Culture 1 (Onom) 1.Annual Juggling Ceremony Values of Culture 2 (Laud) 1.Annual Juggling Ceremony

17. A Tale of Two Kingdoms First Studied by E. Tamref Values of Culture 1 (Onom) 1.Annual Juggling Ceremony 2.Orderly—1 period per year, starting with 1, then 2, 3, etc. Values of Culture 2 (Laud) 1.Annual Juggling Ceremony 2.Orderly—1 period per year, starting with 1, then 2, 3, etc.

18. A Tale of Two Kingdoms First Studied by E. Tamref Values of Culture 1 (Onom) 1.Annual Juggling Ceremony 2.Orderly—1 period per year, starting with 1, then 2, 3, etc. 3.Sequential & Complete— Juggling performances start with all patterns for 0 balls, then 1, 2, 3, etc. Values of Culture 2 (Laud) 1.Annual Juggling Ceremony 2.Orderly—1 period per year, starting with 1, then 2, 3, etc. 3.Sequential & Complete— Juggling performances start with all patterns for 0 balls, then 1, 2, 3, etc.

19. A Tale of Two Kingdoms First Studied by E. Tamref Values of Culture 1 (Onom) 1.Annual Juggling Ceremony 2.Orderly—1 period per year, starting with 1, then 2, 3, etc. 3.Sequential & Complete— Juggling performances start with all patterns for 0 balls, then 1, 2, 3, etc. 4.Individuality— Monistic presentation: 1 performer per ceremony Values of Culture 2 (Laud) 1.Annual Juggling Ceremony 2.Orderly—1 period per year, starting with 1, then 2, 3, etc. 3.Sequential & Complete— Juggling performances start with all patterns for 0 balls, then 1, 2, 3, etc. 4.Complementarity— Dualistic presentation: 2 performers per ceremony

20. The Pact 1400 C.E. In the first year of the new century when the kings of Onom and Laud each decreed the annual juggling period to be 1, a peace treaty was signed. To strengthen this new union, the pact was to be celebrated each year at a banquet where each kingdom would contribute a juggling performance obeying its own principles. However, to symbolize their equal status and mutual regard, each performance must consist of an equal number of juggling patterns.

21. Year One 0 1 2 34 0 1 2 Onom Kingdom Laud Kingdom 0 1

22. Period-1 # of Balls: 01234 # of Patterns:11111

23. Year Two 0 balls1 ball 2 balls 1 pattern3 patterns5 patterns 3 balls 4 balls 7 patterns 9 patterns

24. Year Two Options (patterns with ball-counts 0-4) 001120022231 1340 04 334224 51 15 6006 4453 35 62 2671 17 80 08

25. Year Two—Onom Performer 001120022231134004 334224 51 15 600644 53 35 62 26 71 17 80 08

26. Year Two—Luad Performers 001120022231134004 334224 51 15 6006 Performer 1: Performer 2:

27. Period-2 Patterns per ball are odd numbers A balanced juggling performance: (1+3+5+7+9) = 25 = (1+3+5) + (1+3+5+7) Recall: (the sum of the first n positive odds) = n 2 So: = Onom PerformerLaud Performers

28. Question Will this harmonious arrangement continue indefinitely for the Kingdoms of Laud and Onom? For years 3 and beyond, as the sanctioned periods continually increase by one, can joint ceremonies be planned so that each abides by their own rules and each presents the same number of juggling patterns?

29. Period-2 (again) via initial & middle sums A balanced juggling performance: (1+3+5) + (1+3+5+7) = 25 = (1+3+5+7+9) Subtracting the initial sum (1+3+5) yields: Initial sum = Middle sum (1+3+5+7) = 16 = (7+9)

30. Period-3 Juggling Patterns 0 balls1 ball2 balls… 1 7 19

31. Period-1 # of Balls: 01234 # of Patterns:11111 Period-2 # of Balls: 01234 # of Patterns:13579 Period-3 # of Balls: 01234 # of Patterns:17193761

32. Sequence: 1 719376191 … Sums: 1  8  27  64  125 … 1 3  2 3  3 3  4 3  5 3 … Euler’s Theorem There are no solutions in positive integers a, b, & c to the equation: Period-3

33. Hence… The future of the “Two Kingdoms” is decided by number theory

34. Number Theory T.F.A.E.: 1. 2. 3. For the specific sequences of the form (initial sum) = (initial sum) – (initial sum) (initial sum) = (middle sum)

35. Conclusion Theorem 5: (Graham, et. al., 1994) The number of period-n juggling patterns with fewer than b balls is. Theorem 6: T.F.A.E.: 1. The monistic and dualistic sequential periodic juggling pact can not be satisfied for years 3, 4, 5, … 2. F.L.T.

36. F.L.T. “ It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain. ” Fermat/Tamref Conclusion: “ Add one more to your list of applications of F.L.T. ”

38. Last Thread: Excel spreadsheet explorations of initial and middle sums while juggling the modulus & topics for undergraduate research

39. Initial Sums = Triangular Numbers

41. Initial Sums = First Powers

42. Initial Sums = Squares

43. Initial Sums = Cubes

44. Initial Sums = Fourth Powers

45. Modular Juggling & Juggling with the Modulus

46. Modulus 2 Pattern for Cubic I.S.

47. Modulus 3 Pattern for Cubic I.S.

48. Modulus 4 Pattern for Cubic I.S.

49. Other Mathematical Questions 1.Sequence compression (I.S. seq.)  (base seq.)  (generating seq.)

50. (see Excel)

51. Generating sequence behind the base sequence {1} {1,1} {1,2} {1,6,6} {1,14, 36, 24} HW: What explicit formula derives these generating sequences?

52. Hint—Difference operators

54. Triangular, Square, Cubic Vary IS and Modulus

55. Other Mathematical Questions 1.Sequence compression (I.S. seq.)  (base seq.)  (generating seq.) 2.Patterns of modularity for sequences and arrays

56. A Related Research Topic Modularity patterns in Pascal’s Triangle: See Gallian’s resource page for Abstract Algebra (from MAA’s MathDL) http://www.d.umn.edu/~jgallian/

59. And what is this pattern? I.S. # Mod If properly discerned, a special case of FLT follows (case n = 3).

60. Other Mathematical Questions 1.Sequence compression (I.S. seq.)  (base seq.)  (generating seq.) 2.Patterns of modularity for sequences and arrays 3.Numerous patterns & properties of IS/MS tables 4.Explicit formula for middle sums of fixed length 5.Distribution of IS = MS matches for triangular, square, cubic, or n th power initial sums (why or why not?) 6.Imaginative historical reconstructions— “What margin indeed would have sufficed?”

62. Modular Juggling with Fermat Stephen Harnish Professor of Mathematics Bluffton University harnishs@bluffton.edu Miami University 36 th Annual Mathematics & Statistics Conference: Recreational Mathematics September 26-27, 2008 Archive of Bluffton math seminar documents: http://www.bluffton.edu/mcst/dept/seminar_docs/

63. Modular Juggling with Fermat

65. Website sources Images came from the following sites: http://www.sciamdigital.com/index.cfm?fa=Products.ViewBrowseList http://www2.bc.edu/~lewbel/jugweb/history-1.html http://en.wikipedia.org/wiki/Fermat%27s_last_theorem http://en.wikipedia.org/wiki/Pythagorean_triple http://en.wikipedia.org/wiki/Juggling

66. Another story-line from the 14 th C Earlier in 14 th C. Onom, there had emerged a heretical sect called the neo-foundationalists. They valued orderliness and sequentiality, but they also had more progressive aspirations—the solo performer’s juggling routine would be orderly and sequential but perhaps NOT based on the foundation of first 0 balls, then 1, 2, etc. These neo-foundationalists might start at some non-zero number of balls and then increase from there. However, they were neo-foundationalists in that they would only perform such a routine with m to n number of balls (where 1 < m < n) if the number of such juggling patterns equaled the number of patterns from the traditional, more foundational display of 0 to N balls (for some whole number N). For how many years (i.e., period choices) were these neo- foundationalists successful in finding such equal middle and initial sums of juggling patterns? (Answer: Only for years 1 and 2).