modular juggling with fermat
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Modular Juggling with Fermat
Stephen HarnishProfessor of MathematicsBluffton Universityharnishs@bluffton.edu
Miami University 36th Annual Mathematics & StatisticsConference:
Recreational MathematicsSeptember 26-27, 2008
Archive of Bluffton math seminar documents:
http://www.bluffton.edu/mcst/dept/seminar_docs/
Modular Juggling with Fermat
2 2 23 4 5 n n na b c
Theorem 1: (Euler) The sequence
has no equal initial and middle sums.
Theorem 2: (Dirichlet) The sequence
has no equal initial and
middle sums.
0(3k)(k+1) +1
k
Classical Results
3
05 ( 2) 5 (2 1) 1
kk k k k
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)
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 .
1 2 3, , , , ,kx x x x
1 2 3k kI x x x x
1,
, 1 2j k j j j kM x x x x
,j kM 1k j 1;k j
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)
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
JugglingHistory• 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/
Juggling Patterns (via Juggling Lab)
Thirteen-ball Cascade
A 30-ball pattern of period-15
named:
“uuuuuuuuuzwwsqr”
using standard
site-swap notation
531
Several period-5, 2-ball patterns
90001 12223 30520 14113
A Story Relating Juggling with Number Theory
A Tale of Two KingdomsFirst Studied by E. Tamref
Values of Culture 1 (Onom)
1. Annual Juggling Ceremony
Values of Culture 2 (Laud)
1. Annual Juggling Ceremony
A Tale of Two KingdomsFirst 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.
A Tale of Two KingdomsFirst 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.
A Tale of Two KingdomsFirst 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
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.
Year One
0
1 2
3 4
0
1
2
Onom
Kingdom
Laud
Kingdom
0
1
Period-1# of Balls: 0 1 2 3 4# of Patterns: 1 1 1 1
1
Year Two
0 balls 1 ball 2 balls 1 pattern 3 patterns 5 patterns
3 balls 4 balls7 patterns 9 patterns
Year Two Options(patterns with ball-counts 0-4)
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
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
Year Two—Luad Performers
00 11 20 02 22 31 13 40 04
00 11 20 02 22 31 13 40 04
33 42 24 51 15 60 06
Performer 1:
Performer 2:
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:
==
25 2 23 4Onom Performer Laud Performers
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?
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)
Period-3 Juggling Patterns
0 balls 1 ball 2 balls…
1 7 19
Period-1# of Balls: 0 1 2 3 4# of Patterns: 1 1 1 1
1Period-2
# of Balls: 0 1 2 3 4# of Patterns: 1 3 5 7
9Period-3
# of Balls: 0 1 2 3 4# of Patterns: 1 7 19 37 61
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: 3 3 3a b c
Period-3
Hence…
The future of the “Two Kingdoms” is decided by number theory
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)
n n na b c n n na c b
0
( 1)n n
kk k
ConclusionTheorem 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.
nb
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/TamrefConclusion: “Add one more to your list of applications of F.L.T.”
Last Thread:
Excel spreadsheet explorations of initial and middle sums
while juggling the modulus
&
topics for undergraduate research
Initial Sums = Triangular Numbers
Initial Sums = Triangular Numbers
Initial Sums = First Powers
Initial Sums = Squares
Initial Sums = Cubes
Initial Sums = Fourth Powers
Modular Juggling &
Juggling with the Modulus
Modulus 2 Pattern for Cubic I.S.
Modulus 3 Pattern for Cubic I.S.
Modulus 4 Pattern for Cubic I.S.
Other Mathematical Questions
1. Sequence compression
(I.S. seq.) (base seq.) (generating seq.)
• (see Excel)
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?
Hint—Difference operators
Hint—Difference operators
Triangular, Square, Cubic
• Vary IS and Modulus
Other Mathematical Questions
1. Sequence compression
(I.S. seq.) (base seq.) (generating seq.)
2. Patterns of modularity for sequences and arrays
A Related Research TopicModularity patterns in Pascal’s
Triangle:• See Gallian’s resource page for
Abstract Algebra (from MAA’s MathDL)
• http://www.d.umn.edu/~jgallian/
And what is this pattern?
I.S. #Mod
If properly discerned, a special case of FLT follows (case n = 3).
Other Mathematical Questions
1. Sequence compression(I.S. seq.) (base seq.) (generating seq.)
2. Patterns of modularity for sequences and arrays3. Numerous patterns & properties of IS/MS tables4. Explicit formula for middle sums of fixed length5. Distribution of IS = MS matches for triangular,
square, cubic, or nth power initial sums (why or why not?)
6. Imaginative historical reconstructions—“What margin indeed would have sufficed?”
Modular Juggling with Fermat
Stephen HarnishProfessor of MathematicsBluffton Universityharnishs@bluffton.edu
Miami University 36th Annual Mathematics & StatisticsConference:
Recreational MathematicsSeptember 26-27, 2008
Archive of Bluffton math seminar documents:
http://www.bluffton.edu/mcst/dept/seminar_docs/
Modular Juggling with Fermat
2 2 23 4 5 n n na b c
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
Another story-line from the 14th C
• Earlier in 14th 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).
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