## Problems of the day

How reliable is the Fermat test?

I have encrupted my PIN using
f(x) = x^{7} mod 4159023203.
The result is 2034980130. What is my PIN?

Find the first six terms of the continued fraction expansion
of pi: [a;b,c,d,e,f,...] Then find the convergents
[a;b], [a;b,c], ... [a;b,c,d,e,f].

Find the continued fraction of the square root of 60.
Find a positive solution of Pell's equation x^{2}
- 60y^{2} = 1.

Find a positive solution of Pell's equation x^{2}
- 97y^{2} = 1.

Past problems

## Schedule

Week 1: June 13—16

Week 2: June 20—23

Week 3: June 27—30

## References

A Friendly Introduction to Number Theory, Joseph Silverman

The Higher Arithmetic, Harold Davenport

Elementary Number Theory, William Stein

1978 paper by Rivest, Shamir, and Adleman

1998 Brin-Page Paper (Google)

Cryptography, Tom Davis

Pi is irrational —
source for David McKay's talk

History of math web site

# Summer Mathematics Program for High School Students, June 2005

## Notes and handouts

Square-Triangular Numbers

Eigenvalues and eigenvectors

Mathematics of Google

## Python code

td: td(n) factors n by trial division

gcd: greatest common divisor

isolve: solve ax + by = c

modpower: a^{k} mod N

growcf: growing continued fractions

Basic encryption

Checking up on the Fermat test

Top row, left to right: Ivy Ivanova, Gleb Kuznetsov,

Suzanne Parker, Courtney Ray, Brendan Hoffman,

Duff Evans, Benjamin Skeen, Megan Howarth, Matt

Clay, Andrew Driggs, Russ Ritchin, Adam Trombly.

Bottom row: Tim Simmons, Maria Bell, Cristina Muñoz,

Mike Shrieve, Isaac Allred, Nathan Yu, David McKinlay,

Duncan Humphrey.