1. Introduction and the Sequences of Fibonacci and Collatz
   An introduction to J and experimentation with number sequences to see patterns. Students are led to a proof of Fibonacci identities and see some unanswered questions about the 3x+1 problem.
2. Exploring Prime Numbers: Computations, Theorems and Conjectures
   Implementation of the sieve of Eratosthenese and empirical experiences related to the Prime number theorem and the twin prime conjecture are explored.
3. Euclidean Algorithm and Linear Diophantine Equations
   Develops solutions to a x + b y = gcd(a,b) and and applies that to a word problem. Dealing with more variables is also considered.
4. Solving Equations and Congruences by Brute Force
   Some congruences are solved by testing all residues. Linear, Quadratic, Cubic and Exponential congruemnces are considered. Solving equations by considering suitable congruences is also considered.
5. Some Experiments with Factoring.
   Experiments with trial division, p-1 and Pollard rho and J's built-in q: aimed at giving some sense of the difficulty of factoring.
6. Some Experiments with Primality Testing.
   Uses trial division, Pepin's test, Miller-Selfridge-Rabin test highlighted by experiments with repunits and random integers. The idea is to give some sense of the relative ease with which primality can be tested, at least probabilistically.
7. RSA Codes.
   Algorithm implementation is considered including text to integer conversion and some code breaking experiments. Goal is to include a popular and important application where the students actually deal both with the algorithm and real text.
8. Patterns with Quadratic Residues.
   Leads to discovery of various properties of quadratic residues!
9. Diophatine Equations: Pythagorean Triples and a Pellian Equation.
   Pythagorean triples are determined using brute force and using circle-line intersections. The intersection technique is also applied to the Pellian equation x^2-3y^2=1.
10. Elliptic Curves and Factoring.
    The arithmetic of elliptic curves is introduced and explored. Application to factoring is considered interactively and then compared to Pollard rho.

These laboratories are written in Plain TeX. Faculty interested in browsing copies in TeX, dvi or pdf format are welcome to request copies: email: reiterc@lafayette.edu. Exchanges welcome.