Mathematics 400c
Homework (due Feb. 18)
A. Hulpke
19) Let p be a prime. Show that a p + b p ≡ (a + b) p (mod p).
Hint: Consider the expansion of (a + b) p according to the binomial theorem.
20) a) Determine a number x such that x2 ≡ 15 (mod 113 ).
b) Show that the equation x2 + 4x + 3 ≡ −1 (mod 7123456789 ) has a solution.
21)
Determine all n such that the last four digits of n5 + n4 + 5n2 + 2n + 7 are zero.
22) a) Show that n13 − n is divisible by 2730 for all n.
b) Show that 13|(270 + 370 ).
c) Determine the order of 3 modulo 10007.
22)
Determine all primes p such that p|(2 p + 1).
23) Show that 1062123847 is not prime, by finding a base for which it is not a pseudoprime.
24)∗ You are asked to design a system to facilitate the programming of a VCR which
assigns a (short) unique number to every possible TV program which a user would have to
type in.
Assume there are (at most) 100 channels, that programs can begin and end at times that
are multiples of 15 minutes, and that the system should permit entering codes for films in
the next 4 weeks in advance.
Describe the way how to decode the program information from a program number.
What percentage of the possible numbers are actually used?
How does your system compare with a commercial version that uses up to 8 digits? Can
you imagine reasons for increasing the length of the number? (See also: K. Shirriff, C.
Welch, A. Kinsman, Decoding a VCR Controller Code, Cryptologia, 16(3), July 1992, pp
227-234, and http://tallyho.bc.nu/˜steve/videoplus.html)
Problems marked with a ∗ are bonus problems for extra credit.