189-235A: Basic Algebra I
Due: Wednesday, September 10.
From the textbook:
Page 7, 1.3, 1.5.
Page 17, 1.14, 1.16, 1.17, 1.21, 1.22.
8. Pascal's triangle modulo 2 is an infinite
triangular array of numbers
consisting entirely of 0's and 1's, defined according to the following rule:
an entry contains 1 if the corresponding entry in
Pascal's triangle is odd, and it contains a 0 if the corresponding
entry in Pascal's triangle is even.
Making as little computation as possible,
write down the first 17 rows of
Pascal's triangle modulo 2. What patterns do you observe? Try formulating
precise conjectures. Can you prove your conjectures?
To learn more about Pascal's triangle modulo 2, see the article
Zaphod Beeblebrox's brain and the fifty-ninth row of
Pascal's triangle by Andrew Granville,
in the American Mathematical Monthly, 99 (1992),
no. 4, pages 318-331.