Homework 11

This assignment is due to my office by 11am on Wednesday, November 19th. Be sure to review the homework guidelines before getting started.

• Create your own public key (your primes should be greater than 100000), where you'll choose to encode using 8 unit blocks . Encode a message of your choice (at least 12 characters in length). Make sure you describe the cleartext to plaintext conversion. Be creative.
• Your arch nemesis, Chris, uses the public key $m=41227987$ and $e=31861031$, and he tells people to encode using 8-unit blocks. You happen to have reliable intelligence, however, that Chris' favorite prime number is 541. Use this to try to decode the following message which you've intercepted on its way to Chris.

18501764, 17524287, 4260339, 10258276, 7904375, 5489559, 39001743, 36111043, 25847617

• Use integration by parts to show $\Gamma(s+1) = s\Gamma(s)$ for every s. Use this result to show $\Gamma(n+1) = n!$ for any positive integer n.

Bonus:

• Use the Euler Product formula for the Riemann Zeta function to prove that $\zeta^2(s) = \sum_{n=1}^\infty \frac{\nu(n)}{n^s}$.