This Saturday’s (9/27) Math Circle lecture by Dr. Bennett Harris on “Computer Data Encryption – Decrypted” promises to be a great combination of number theory and applied math. To warm up the students, Dr. Harris forwarded some problems that I will post in two parts. Feel free to offer solutions in the comments or to just work them on your own. Full solutions will be made available at the next Math Circle.
A solution for each of the following should either give the correct answer, or a technique for determining the answer in reasonable time with the assistance of a calculator.
1. How many prime numbers are there? Prove your answer.
2. What is the largest number you must test to demonstrate that 83 is prime?
3. What is the smallest 5-digit prime? How could you find this number?
4. Is 1234567890 prime? What about 123456789?
5. Assume an alphabet with 26 letters (plus a blank space). A substitution cipher is a one-to-one mapping of this alphabet onto itself. How many such substitution ciphers are there?