PascGalois Activities for a Number Theory Class Kurt Ludwick Salisbury University Salisbury, MD

Developed by Kathleen M. Shannon & Michael J. BardzellKathleen M. ShannonMichael J. Bardzell Salisbury University, Salisbury, MD Support provided by The National Science Foundation award #'s DUE and DUE and- The Richard A. Henson endowment for the School of Science at Salisbury University PascGalois Visualization Software for Abstract Algebra

PascGalois Applications in a Number Theory class? Main idea: Pascal’s Triangle modulo n ( 1-dimensional finite automata) Primary use: Abstract Algebra classes ( Visualization of subgroups, cosets, etc.) 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

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

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

A few examples….. Pascal’s Triangle modulo 12 – rows

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…..

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) These characteristics can be seen under other moduli as well….. 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

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)……

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.

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!)

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.

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

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…

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?

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 2 2 M, where M is a product of primes not equal to 2 (same meaning for “M” throughout…..) Size 7 triangles begin in rows numbered 2 3 M Size 15 triangles begin in rows numbered 2 4 M …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 2 k -1, where 2 k is the greatest power of 2 that divides the row number

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 p 2 -1 triangles begin in rows numbered p 2 M …within Pascal’s Triangle (modulo p), p an odd prime, the size of a solid red triangle will be p k -1, where p k 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.

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?

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

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: o The row number, r o From row r, the horizontal position of each non-red (non-zero) entry As an example, we will consider row r=32 of Pascal’s Triangle modulo 5. So, r= The non-zero locations in this row are: 0, 1, 2, 5, 6, 7, 25, 26, 27, 30, 31 and 32.

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…..

Activity #2 – Lucas Correspondence Theorem The Lucas Correspondence Theorem: Let p be prime, and let k, r be positive integers with base p digits k i, r i, respectively. That is, Then, Notice: iff r i < k i, which is why an entry in the k th 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.

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

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

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.

Activity #2 – Lucas Correspondence Theorem Example: Pascal’s Triangle (mod 7), row 23 Pascal’s Triangle (mod 7) Rows 0-27 r = 23 = 32 7 Nonzeros : k = 0, 1, 2, 7, 8, 9, 14, 15, 16, 21, 22, 23

Developed by Kathleen M. Shannon & Michael J. BardzellKathleen M. ShannonMichael J. Bardzell Salisbury University, Salisbury, MD Support provided by The National Science Foundation award #'s DUE and DUE and- The Richard A. Henson endowment for the School of Science at Salisbury University PascGalois Visualization Software for Abstract Algebra THANK YOU!