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Assignment 8: Due on Friday, November 22
Warm-up problems (some of them are difficult, though!)
(1) Use the fact that ex^ = 1 + x + x^2 /2! + x^3 /3! + x^4 /4! + · · · for all real x to show that nn^ = O(n!en). (2) Show that if C > 1 then Cn^ = o(n!). (3) Show that log(n) = o(nc) and if c, d > 1 and m ≥ 0 then (logc(n))m^ = O((logd(n))m). (4) Let f 1 , g 1 , f 2 , g 2 : N → (0, ∞). Show that if f 1 (n) = O(g 1 (n)) and f 2 (n) = O(g 2 (n)) then f 1 (n) + f 2 (n) = O(g 1 (n) + g 2 (n)). (5) Let f, g, h : N → (0, ∞). Show that if f (n) = O(g(n)) and g(n) = O(h(n)) then f (n) = O(h(n)). (6) Let S ⊆ N and suppose that |{n ≤ x : n ∈ S}| = O((log(x))d) for some d ≥ 1. Show that for each ϵ > 0 the sum X
s∈S
1 /sϵ
converges. (7) Given two functions f, g : N → (0, ∞), we say that f (n) is asymptotic to g(n) if f (n)/g(n) → 1 as n → ∞ and we write f (n) ∼ g(n). If you know about Riemann integration, show that we have the following inequalities for n ≥ 2:
log(n − 1) ≤
X^ n
i=
1 /i ≤ 1 + log(n).
Use this to show that (1 + 1/2 + 1/3 + · · · + 1/n) ∼ log(n + 1). (8) Use the above estimates to show that 1 /(n + 1) + · · · + 1/(2n) ∼ log(2). (9) The prime number theorem is a famous theorem that says that if π(x) denotes the number of (positive) prime numbers p that are less than x then π(n) ∼ n/ log(n). Show that the prime number theorem shows that there is some positive integer N such that there is a prime number in the interval {n, n + 1,... , 2 n} whenever n ≥ N. (10) Show that given integers m, n > 1, each with at most k binary digits, one can compute gcd(m, n) with O(kd) operations for some d ≥ 1 that is independent of k. (You can use basic results about complexity of addition, subtraction, and the division algorithm.) (11) Consider the following algorithm for sorting a list of n distinct integers a 1 ,... , an. For i = 1,... , n − 1, we start at i = 1 and we compare ai and ai+1. It ai < ai+1 then leave ai and ai+1 unchanged and increment i by 1; otherwise, exchange ai and ai+1 (so ai+1 is now ai and ai is now ai+1) and again increase i by 1. After we finish step n − 1 we have a new list 1
b 1 ,... , bn, which is a rearrangement of our original list. If this new list is the same as our original list then stop and return this list; otherwise, run the procedure above again. For example, if we have n = 4 and our original list is 4, 10 , 9 , 1, then after the first iteration we obtain the new list 4, 9 , 1 , 10. Then we do it again and obtain the list 4, 1 , 9 , 10, and then we do it again and we get 1, 4 , 9 , 10. Finally, we do it again and the list is unchanged so we stop and return 1, 4 , 9 , 10. Another example is with the list 5 , 4 , 3 , 2 , 1; on the first iteration, we get the new list 4, 3 , 2 , 1 , 5; then we get 3, 2 , 1 , 4 , 5; then we get 2, 1 , 3 , 4 , 5; and then we get 1, 2 , 3 , 4 , 5, which stays the same when we apply the list so we return 1, 2 , 3 , 4 , 5 Show that the above procedure always sorts the integers a 1 ,... , an in order from smallest to largest and that for a list of n elements this algorithm requires O(n^2 ) operations. (12) Let n be a positive integer and let d(n) denote the number of elements in { 1 , 2 ,... , n} that divide n. Show that d(n) = O(
n). (13) Define a sequence an as follows. We take a 1 = 1 and for n ≥ 2 we define an = a^2 n− 1 + 1. Show that an = o(2^2 n ) and that c^2 n = o(an), where c = 2^1 /^4. (14) Let n ≥ 1. Show that X
d|n
ϕ(d) = n,
where ϕ(m) is Euler’s totient function and the sum runs over all posi- tive divisors of n. (Hint: Use the unique factorization of n = pa 11 · · · pa s s. Then the divisors of n are pb 11 · · · pb ss with 0 ≤ bi ≤ ai for i = 1,... , s. Recall that we showed ϕ(pb 11 · · · pb ss ) = ϕ(pb 11 ) · · · ϕ(pb ss ). Thus X
d|n
ϕ(d) =
X^ a^1
b 1 =
X^ as
bs=
ϕ(pb 11 ) · · · ϕ(pb ss ),
which can be written as X^ a^1
b 1 =
ϕ(pb 11 )
X^ as
bs=
ϕ(pb s^1 )
Now show that ϕ(pk) = pk(1 − 1 /p) if k ≥ 1 and is 1 otherwise and use that to show that this sum is pa 1 1 · · · pa s s= n.) (15) Find q(x), r(x) ∈ Q[x] with deg(r(x)) < 3 such that x^5 + x^3 + 1 = q(x)(x^3 − 2) + r(x). (16) Let p(x) be a polynomial of odd degree in R[x]. Show that p(x) has a real root. (Hint: use the intermediate value theorem.) (17) Question 1 of the assignment asks you to show that a monic polynomial P (x) of degree n has at most n roots in a field F. Show that this is
(2) Let G be a finite group and suppose that for every n ≥ 1 there are at most n elements x ∈ G that satisfy xn^ = 1. Show that G is cyclic. (Hint: Let Gn denote the set of elements x ∈ G whose order is exactly n, that is xn^ = 1 and xi^ ̸= 1 for i = 1,... , n−1. Show that |Gn| ≤ ϕ(n). Show that |G| is exactly the sum
P
d|N |Gd|^ when^ |G|^ =^ N^. Then use a warm-up exercise to show that N =
P
d|N |Gd| ≤^
P
d|N ϕ(d) =^ N^. Conclude that |Gd| = ϕ(d) whenever d | N .) (3) Let F be a finite field. Use the preceding exercises of the assignment to show that F ∗, the set of nonzero elements of F is a cyclic group of order |F | − 1 and conclude that (Z/pZ)∗^ is a cyclic group of order p − 1 when p is prime.
