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## Dr. Bennette Harris – Problem Set 2

Here is the continuation of Dr. Bennett Harris’ problems to warm us up before his lecture on September 27th. Congratulations to Dominic for being the first to answer the previous set of problems in the comments. Another happy discovery is the fact that WordPress, which hosts this site, supports the use of $\LaTeX$ code!

### Problems

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.

6. Crack this secret message: “uifsfaxjmmacfabmnptuaopaqsppgtaupebz”

7. Code the alphabet a=0, b=1, …, z=25, space=26. We will encrypt each letter of the alphabet by mapping it to the letter 3 farther along in the alphabet. Thus, a maps to d, b maps to e, and so forth. The message “a bat is fat” would be encrypted as “dcedwclvciodw” in this scheme. Find a mathematical function f(x) such that y=f(x) gives the correct encryption for every value x = 0,1,2,…,26. Test your function by encrypting “the toy boat floats”.

8. The purpose of these two questions is to provide points of comparison for the questions that follow:
How many particles are there in the known universe?
How many microseconds have elapsed since the beginning of the universe?

9. How long (approximately will do) would it take to calculate $10000^{1234567890}$ at 1,000,000 calculations per second? How many digits does the answer have? If p is 200 digits long, and if q is 200 digits long, about how many digits are there in pq?

10. The number $2^{521}$ – 1 is known to be prime. Estimate how long it would take a computer to demonstrate this by repeated division, at 1,000,000 divisions per second.