HW 4: due Friday 9/17

READ: Sections 2.3, B3 Ex.4 (on Friday we will learn how to decrypt the Vigenere cipher on the computer).
BOOK PROBLEMS: Section 2.13 #16, 3.11 #3a,#4, 3.12 #5.
OTHER PROBLEMS:
A. Find two numbers n and n+35 which have no primes between them.

B. Show that 207=x^2+y^2+z^2 has no solutions if x,y,z are integers. Hint: look mod 8.

C. Multiple encryption: Pick two numbers m and n.
What is the period if you encrypt a message first with a Vigenere cipher of period m and then encrypt the ciphertext with a Vigenere cipher of period n?
Your answer should depend on m and n.

D. For an alphabet of N=27 letters, find the number of shift ciphers, affine ciphers, substitution ciphers, and vigenere ciphers with period 20.
Do the same for N=30.

E. What is the dot product of (1,0,0,....0) with A0?
What is the dot product of (1,1,1,....1) with A0? What do these numbers represent?

Bonus:
1. Find formulae for problem D in the case of an arbitrary number N of letters.
2. Using data on page 21, find which vector Ai is closest to the vector A0 (other than A0 itself). What is the angle between them?