Modular Juggling with Fermat

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

2. Modular Juggling with Fermat

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

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 15th 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/

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 90001 12223 30520 14113

15. A Story Relating Juggling with Number Theory

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

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

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

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

20. The Pact1400 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. Onom Kingdom Laud Kingdom Year One 0 0 0 2 1 1 1 2 3 4

22. Period-1 # of Balls: 0 1 2 3 4 # of Patterns: 1 1 1 1 1

23. Year Two 0 balls 1 ball 2 balls 1 pattern3 patterns5 patterns 3 balls 4 balls 7 patterns9 patterns

24. Year Two Options(patterns with ball-counts 0-4) 00 11 20 0222 31 13 40 04 33 42 24 51 15 60 06 44 53 35 62 26 71 17 80 08

25. Year Two—Onom Performer 00 11 20 02 22 31 13 40 04 33 42 24 51 15 60 06 44 53 35 62 26 71 17 80 08

26. Year Two—Luad Performers Performer 1: 00 11 20 02 22 31 13 40 04 Performer 2: 00 11 20 02 22 31 13 40 04 33 42 24 51 15 60 06

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) = n2 So: = = Onom Performer Laud 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 balls 1 ball 2 balls… 1 7 19

31. Period-1 Period-2 # of Balls: 0 1 2 3 4 # of Patterns: 1 1 1 1 1 # of Balls: 0 1 2 3 4 # of Patterns: 1 3 5 7 9 Period-3 # of Balls: 0 1 2 3 4 # of Patterns: 1 7 19 37 61

32. Sequence: 1 7 19 37 61 91 … Sums: 1  8  27  64  125 … 13 23  33 43 53 … 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.: • The monistic and dualistic sequential periodic juggling pact can not be satisfied for years 3, 4, 5, … • 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 themodulus& topics for undergraduate research

39. Initial Sums = Triangular Numbers

40. 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 • Sequence compression (I.S. seq.)  (base seq.)  (generating seq.)

50. (see Excel)