Hello, if you have any need, please feel free to consult us, this is my wechat: wx91due
Math 115B Homework 1
1) The following problem is open, meaning no one knows a solution: Let n 2 Z with n > 1. If (n)|n 1, then n is prime. Prove the following weaker result: Let n 2 Z with n > 1. If (n)|n1, then n is a product of distinct prime numbers.
2) Find (n) and ⌧ (n) for n = 64, 105, 2592, 4851, 111111, and 15!.
3) a) Characterize the positive integers n for which ⌧ (n) is odd.
b) Characterize the positive integers n for which (n) is odd.
4) Let k 2 Z with k > 1. Prove that the equation ⌧ (n) = k has infinitely many solutions n.
5) Let n 2 Z with n > 0. a) Show that ⌧ (n) 2 pn. b) Show that ⌧ (n) ⌧ (2n 1).
6) a) Let n 2 Z with n > 0. Show that P d|n,d>0 1 d = (n) n . b) Let n be a perfect number. Show that P d|n,d>0 1 d = 2.
7) Let n 2 Z with n > 0. The number n is said to be superperfect if ((n)) = 2n.
a) Prove that 16 is superperfect.
b) Prove that if 2p 1 is a Mersenna prime, then 2p1 is superperfect.
c) Prove that if 2a is superperfect, then 2a+1 1 is a Mersenne prime. (Note: it turns out that all even superperfect numbers are of the form 2p1 where 2p 1 is a Mersenne prime. No odd superperfect numbers are known)
8) How dicult was this homework? How long did it take?