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PascGalois Activities for a Number Theory Class

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PascGalois Activities for a Number Theory Class

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  1. PascGalois Activities for a Number Theory Class Kurt Ludwick Salisbury University Salisbury, MD http://faculty.salisbury.edu/~keludwick

  2. 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

  3. PascGalois http://www.pascgalois.org • Primary use: Abstract Algebra classes ( Visualization of subgroups, cosets, etc.) • Main idea: Pascal’s 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

  4. A few PascGalois examples….. Pascal’s Triangle modulo 2 – rows 0 - 64 Even numbers: red Odd numbers: black

  5. A few examples….. A few examples….. Pascal’s Triangle modulo 5 – rows 0 - 50 Colors correspond to remainders Notice “inverted” red triangles, as were also seen in the modulo 2 triangle

  6. A few examples….. A few examples….. Pascal’s Triangle modulo 12 – rows 0 - 72

  7. 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 Pascal’s 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 Pascal’s Triangle (modulo 2) These characteristics can be seen under other moduli as well…..

  8. 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 Pascal’s 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 Pascal’s Triangle (modulo 5)

  9. 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

  10. Activity #1 – Inverted triangles Questions: 1. Are the solid triangles always inverted? Suppose not… then, the following must occur somewhere within Pascal’s Triangle (modulo n)for some X, 0 < X < 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)……

  11. Activity #1 – Inverted triangles Questions: 1. Are the solid triangles always inverted? Suppose not… then, the following must occur somewhere within Pascal’s Triangle (modulo n)for some X, 0 < X < n-1: Thus, by contradiction (and using properties of modular arithmetic), no “right-side-up” triangles of size 3 (or greater) can occur.

  12. 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!)

  13. 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.

  14. 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

  15. 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…

  16. 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?

  17. 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 Pascal’s 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 Pascal’s 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

  18. 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 Pascal’s 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 Pascal’s 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.

  19. 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: Pascal’s 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?

  20. 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

  21. Activity #2 – Lucas Correspondence Theorem Instructions: • Choose a prime number, p. Use PascGalois to generate Pascal’s 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 r=32of Pascal’s Triangle modulo 5. So, r=1125. The non-zero locations in this row are: 0, 1, 2, 5, 6, 7, 25, 26, 27, 30, 31 and 32.

  22. Activity #2 – Lucas Correspondence Theorem Observation (after a few examples): The kth 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…..

  23. 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 < 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.

  24. Activity #2 – Lucas Correspondence Theorem Following the same example, we have….

  25. Activity #2 – Lucas Correspondence Theorem Following the same example, we have….

  26. 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 Lucas’s Theorem.

  27. Activity #2 – Lucas Correspondence Theorem Example: Pascal’s Triangle (mod 7), row 23 Pascal’s Triangle (mod 7) Rows 0-27 r = 23 = 327 Nonzeros: k = 0, 1, 2, 7, 8, 9, 14, 15, 16, 21, 22, 23

  28. 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!