PascGalois Activities

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PascGalois Activities for a Number Theory Class
Kurt Ludwick Salisbury University Salisbury,
MD http//faculty.salisbury.edu/keludwick
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PascGaloisVisualization Software for Abstract
Algebra http//www.pascgalois.org
Developed by Kathleen M. Shannon Michael J.
Bardzell Salisbury University, Salisbury,
MD Support provided by The National Science
Foundation award 's DUE-0087644 and
DUE-0339477 -and-The Richard A. Henson endowment
for the School of Science at Salisbury University
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PascGalois http//www.pascgalois.org

• Primary use Abstract Algebra classes (
  Visualization of subgroups, cosets, etc.)

• Main idea Pascals Triangle modulo n (
  1-dimensional finite automata)

Applications in a Number Theory class?

• A few class objectives
• Properties of modular arithmetic
• Properties of binomial coefficients
• Inductive reasoning
• Observing a pattern
• Clearly stating a hypothesis
• Proof
• Significance of prime factorization

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A few PascGalois examples..
Pascals Triangle modulo 2 rows 0 - 64
Even numbers red Odd numbers black
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A few examples..
A few examples..
Pascals Triangle modulo 5 rows 0 - 50
Colors correspond to remainders Notice
inverted red triangles, as were also seen in
the modulo 2 triangle
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A few examples..
A few examples..
Pascals Triangle modulo 12 rows 0 - 72
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Activity 1 Inverted triangles
Discovery activity (ideally) best suited as
interactive assignment in a computer lab (can
also work as an out-of-class assignment, with
detailed instructions)
Notice the solid triangles with side length at
least 3 within Pascals Triangle (modulo 2).
What do we observe about them?

• They are all red

• They are all upside down (Longest edge is
  at the top)

• Their sizes vary throughout the interior of
  Pascals Triangle (modulo 2)

These characteristics can be seen under other
moduli as well..
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Activity 1 Inverted triangles
These characteristics can be seen under other
moduli as well..
Notice the solid triangles with side length at
least 3 within Pascals Triangle (modulo 5).
What do we observe about them?

• They are all red

• They are all upside down (Longest edge is
  at the top)

• Their sizes vary throughout the interior of
  Pascals Triangle (modulo 5)

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Activity 1 Inverted triangles Questions
1. Are the solid triangles always inverted?
2. Are the solid triangles always red?
3. Find the size (defined as length of the top
row) of each triangle as a function of the
row number in which its top row appears
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Activity 1 Inverted triangles Questions
1. Are the solid triangles always inverted?
Suppose not then, the following must occur
somewhere within Pascals Triangle (modulo
n)for some X, 0 lt X lt n-1
where none of the entries labeled ? may be
equal to X
We can see that certain of the ? entries must
be 0 (implying X is not 0)
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Activity 1 Inverted triangles Questions
1. Are the solid triangles always inverted?
Suppose not then, the following must occur
somewhere within Pascals Triangle (modulo
n)for some X, 0 lt X lt n-1
Thus, by contradiction (and using properties of
modular arithmetic), no right-side-up triangles
of size 3 (or greater) can occur.
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Activity 1 Inverted triangles Questions
2. Are the solid triangles always red?
Yes, by a similar argument to have an inverted
triangle of a single color, X, it would be
necessary to have
which implies X 0 , or red.
(The standard coloring scheme in PascGalois is to
have red designate the zero remainder. This can
be customized, of course!)
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Activity 1 Inverted triangles Questions
3. Find the size (defined as length of the top
row) of each triangle as a function of the
row number in which its top row appears
Size 3.
Row 4
Row 12
Row 20
Row 28
etc.
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Activity 1 Inverted triangles Questions
3. Find the size (defined as length of the top
row) of each triangle as a function of the
row number in which its top row appears
Size 7.
Row 8
Row 24
Row 40
Row 56
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Activity 1 Inverted triangles Questions
3. Find the size (defined as length of the top
row) of each triangle as a function of the
row number in which its top row appears
Size 15.
Row 16
Row 48
Next 80, 112, 144, etc
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Activity 1 Inverted triangles Questions
3. Find the size (defined as length of the top
row) of each triangle as a function of the
row number in which its top row appears
Size 31.
Row 32
..next?
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Activity 1 Inverted triangles Questions
3. Find the size (defined as length of the top
row) of each triangle as a function of the
row number in which its top row appears
To answer this question completely, one must
use the prime factorization of the row number.

• In Pascals Triangle (modulo 2)
• Size 3 triangles begin in rows numbered 22M,
  where M is a product of primes not equal to 2
  (same meaning for M throughout..)

• Size 7 triangles begin in rows numbered 23M

• Size 15 triangles begin in rows numbered 24M

and so on in general, within Pascals Triangle
(modulo 2), the size of a solid red triangle
starting on a given row will be 2k-1, where
2k is the greatest power of 2 that divides
the row number
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Activity 1 Inverted triangles Questions
3. Find the size (defined as length of the top
row) of each triangle as a function of the
row number in which its top row appears
Generalizing to Pascals Triangle (modulo p), for
prime p

• Size p-1 triangles begin in rows numbered pM,
  where M is a product of primes not equal to p

• Size p2-1 triangles begin in rows numbered p2M

within Pascals Triangle (modulo p), p an odd
prime, the size of a solid red triangle will
be pk-1, where pk is the greatest power of p
that divides the row number
To come up with this solution, students must get
used to thinking about integers in terms of their
prime factorization.
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Activity 1 Inverted triangles Questions
3. Find the size (defined as length of the top
row) of each triangle as a function of the
row number in which its top row appears
Example Pascals Triangle (modulo 5)
Red triangles of size 4 begin on rows 5, 10, 15,
and 20.
A red triangle of size 24 begins on row 25.
More triangles of size 4 begin on rows 30, 35,
40 and 45..
Guess what happens on row 50?
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Activity 1 Inverted triangles

• Summary
• Gives students experience working with the
  PascGalois software
• Provides a few easy proofs involving
  properties of modular arithmetic
• Introduces (or reinforces) the idea of thinking
  of the natural numbers in terms of their prime
  factorizations

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Activity 2 Lucas Correspondence Theorem
Instructions

• Choose a prime number, p. Use PascGalois to
  generate Pascals Triangle modulo p.

• Choose a row in this triangle. Let r denote
  the row number you choose.

• Write out each of the following in base p
• The row number, r
• From row r, the horizontal position of each
  non-red (non-zero) entry

As an example, we will consider row r32of
Pascals Triangle modulo 5. So, r1125. The
non-zero locations in this row are 0, 1, 2,
5, 6, 7, 25, 26, 27, 30, 31 and 32.
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Activity 2 Lucas Correspondence Theorem
Observation (after a few examples) The k th
position in row r is nonzero (mod p) iff each
digit of k is less than or equal to the
corresponding base p digit of r.
This is an observation in the direction of what
is known as the Lucas Correspondence Theorem..
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Activity 2 Lucas Correspondence Theorem
The Lucas Correspondence Theorem
Let p be prime, and let k, r be positive integers
with base p digits ki, ri, respectively. That
is,
Then,
Notice iff ri lt ki, which is
why an entry in the kth position is zero (mod
p) iff at least one of its base p digits is
greater than the corresponding digits of the
row number, r.
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Activity 2 Lucas Correspondence Theorem
Following the same example, we have.
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Activity 2 Lucas Correspondence Theorem
Following the same example, we have.
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Activity 2 Lucas Correspondence Theorem
Following the same example, we have.
Note for any other value of k, one of the three
factors (and thus the product) in the right-hand
column is zero, corresponding to a binomial
coefficient that is congruent to 0 (mod 5), as
per Lucass Theorem.
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Activity 2 Lucas Correspondence Theorem
Example Pascals Triangle (mod 7), row 23
Pascals Triangle (mod 7) Rows 0-27
r 23 327
Nonzeros k 0, 1, 2, 7, 8, 9, 14, 15, 16, 21,
22, 23
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PascGaloisVisualization Software for Abstract
Algebra http//www.pascgalois.org
Developed by Kathleen M. Shannon Michael J.
Bardzell Salisbury University, Salisbury,
MD Support provided by The National Science
Foundation award 's DUE-0087644 and
DUE-0339477 -and-The Richard A. Henson endowment
for the School of Science at Salisbury University
THANK YOU!