# The Lucas Correspondence Theorem

### Introduction

Given a prime number *p* and integers *r*, *k*
such that *r* > 0 and 0 <= *k* <= *r*, we will write the base *p*
representations of *r* and *k* as follows:

where *m* is chosen such that *p*^{m} <= r < *p*^{m+1}.
We will consider row *r* of Pascal's Triangle modulo *p*.
As an example, let *p* = 7 and *r* = 22; that is, we
will consider the 22^{nd} row of Pascal's Triangle modulo 7:

If we write the row number and the locations of the
non-zero entries, all in base 7 (including leading zeros,
if necessary), we get the following table:

From examples such as this one, we wish to infer a method of
predicting the nonzero entries of row *n* of
Pascal's triangle modulo *p*.

### Activity Instructions

- Following the above example, create similar tables for rows
14, 16, 18 and 20 of Pascal's triangle
modulo 7. In each case, what is the relationship between the base
*p* digits of the row number and
the base *p* digits of the locations of the nonzero entries?
When using this applet to graph Pascal's triangle modulo *n* you
will need to

- Change the n = box to your modulus.
- Place a 1 in one of the seed boxes and remove any entry in the other seed box.

Also, remember that the divider between the triangle and the color scheme
is moveable.

- For a few different values of
*k* from row *r* = 18 of the
(mod 7) triangle, calculate the binomial coefficient
for each *j*. In each case, compare your results to the value of
Write out your results.
- For each of the examples you considered in
# 2, look for a connection between the values of
mod 7 and the value of
mod 7. In particular, consider the product
Keep in mind that when a < b,
For example: If
*r* = 18 = 24_{7} and *k* = 10 = 13_{7}, then:
Come up with a conjecture about the connection between
- Test your conjecture for some other prime moduli
*p* and rows *r*.
(So far we’ve only really considered
*p* = 7 and *r* = 18.) Consider at least two other values of
*p*; for each of these, repeat the procedures
described in #1 and #2 above for at least two or three different rows
of Pascal's triangle modulo *p*.
Does your conjecture from # 3 seem to hold up? Do you think it is true in general?
- Look up the "Lucas Correspondence Theorem" on the internet.
(For example, you can find it on
http://mathworld.wolfram.com.)
Compare what you find to what you came up with in #3–4 above.
- Describe how the Lucas Correspondence Theorem reveals the locations of the zero versus
non-zero entries of Pascal's triangle modulo a prime
*p*.