Exercises

• Chapter 1
  1. The Sieve of Eratosthenes is a method of producing primes within a given range which goes like this ... Use the Sieve to find all the primes less than 30. How does this relate to the fifth proof given for the infinity of primes?
  2. Use the Euler identity Pi (1-p^-1)^-1 = Sum 1/n = infinity, (see proof #4) to prove sum_n=1 mu(n)/n = 0. [Remember the definition of mu(n) = (-1)^k if n is the (square free) product of k distinct primes, mu(1) = 1 and otherwise mu(n) = 0.]
  3. Modify Euclid's proof to show there are infinity many primes of the form 4n+3. Show why this proof fails to show that there are infinitely many primes of the form 4n+1. [but see chap 4].
  4. By considering prime divisors of the form n^2+2, prove there are infinitely many primes of the form 8k+3.
  5. Show there are infinitely many primes of the form 6k+1.
  6. Prove that the odd divisors of n^4+1 are of the form 8k+1. Prove there are infinitely many primes of the form 8k+1.
  7. if p is prime and 3p+1 is a square, then p=5.
  8. The product of n consecutive positive integers is divisible by n!.
  9. if p is prime (p-1)! congruent to p-1 mod (1+2+3+..(p-1))
  10. if gcd(a,n) = gcd(a-1,n-1)=1 then 1+a+a^2+...+a^(phi(n)-1) congruent 0 mod n
  11. old prime divisors of n^2+1 are of the form 4k+1.
  12. old prime divisors of n^4+1 are of the form 8k+1.
  13. old prime divisors (other than 3) of n^2+n+1 are of the form 6k+1.
  14. The second proof is based on the fact that the Fermat numbers are relatively prime. Show if a_0 is odd ≥ 3, then a_n+1 = (a_n)^2 -2 is also a relately prime sequencee.
  15. Suppose p_1 ... p_k are the only primes, N is the product of the p_i's, Q_i = N/p_i and S = sum{Q_i, i=1..k}. Show that no prime divides S.
  16. Suppose p_1 ... p_k are the only primes, N is the product of the p_i's, show gcd(N, N-1) = 1 and there are infinitely many primes.
• Chapter 2
  1. Give asymtotically tight bounds on the the following summations. sum{k^2, k=1..n}; sum{log^s k, k=1..n}; sum{k^tlog^s k, k=1..n} (s, t are non-negative integers)
  2. Show the binomial coefficient 2n+1 choose n ≥ 2^(2n).
  3. Ishikawa proved in 1934, pi(xy> pi(x) + pi(y) for large enough x, y show how Bertrands Postulate follows from this result.
• Chapter 4
  1. Represent 113, 229, 373 as the sum of two squares.
  2. Find five more primes of the form n^2+(n-1)^2 the first two are 5 = 1^2 + 2^2 and 13 = 2^2+3^2.
  3. prove n can be written as x^2 - y^2 iff n is the product of two numbers which are both odd or both even.
  4. Show 45 is the smallest integer that can be written as the sum of two squares in two different ways.
  5. Show 2^(2n+1) = (2^(2n-k) + 2^(k-1))^2 - (2^(n-k) - 2^(k-1))^2 so there are integers that can be expressed as the difference of two squares n different ways.
  6. If q and p are distinct primes then p^(q-1)+q^(p-1) = 1 mod pq.
  7. If p is prime and a^p = b^p mod p, the a^p = b^p mod p^2.
  8. Prove that 2^(2^n)-1 has at least n distinct prime factors.
  9. Prove that n! is never a perfect square (for n &ge 2).
  10. If p is of the form 4k+3 then ((p-1)/2)! or (p-1)/2 = -1 mod p.
• Chapter 6
  1. Prove the m-th root of a is rational if and only if a = b^m (a, b integers).
  2. sqrt(2) is an example of a rational to a rational power that is irrational, e^r is an example of an irrational to a rational power, which is irrational. Find an example of an irrational to an irrational power which is rational. [hint let a = sqrt(2), consider a^a and (a^a)^a.]
  3. Show that sometimes e^b is irrational and sometimes it is rational as b ranges over the irrationals. [hint chap 15]
  4. Show the integral from 0 to infinity of sin x/x = pi/2.
• Chapter 7
  1. If A, B, C and D are four point in the plane |AD||BC| ≤ |BD||CA|+|CD||AB|
  2. Using |z_1*z_2| = |z_1||z_2| express the product (a_1^2+b_1)^2(a_2^2+b_2)^2 as the sum of two squares.
  3. From the formula |PQ|^2 = (P-Q)^2 derive the cosine formula for the triangle 0PQ.
  4. If A, B, and C are non-colinear points, any vector P may be expressed in terms of A, B, and C as P = xA + yB + zC where x + y + z = 1. [This requires one to be in the plane.]
• Chapter 10
  1. If all vertices have degree d and all faces have degree r in a plane graph find all possible d and r. [These lead to the regular polyhedra.]
• Chapter 13 (perhaps chapter 7?)
  1. can one divide a cube into congruent and disjoint tetrahedra?
  2. how many d-simplices are needed to ``triangulate'' a d-hypercube?
  3. an unfolding of a 3d convex polytope is obtained bu cutting the polytope along some of the edges (necessarily a spanning tree of the edge graph) and flattening the boundary along the remaining edges. Does ever convex polytope have a simple unfolding (one that does not intersect itself)?
  4. Call a facet of a d-dim polytope degenerate if it has more than d vertices. How many degenerate facets can an n-vertex polytope have in the worst case?
• Chapter 15
  1. if |A| ≤ |B| and if A is nonempty, then there exists a mapping f of B onto A.
  2. Let A be countable, the set [A]^n = {S subset A: |A| = n} is countable for all positive integers n.
  3. Where does the proof of theorem 2 break down if the decimal expansion is broken down digit by digit.
  4. Produce and explicit bijection |N -> |Z and show it is such.
  5. Prove that an intersection of countable sets is at most countable.
  6. Show the irrationals are uncountable.
  7. Prove the set of all algebraic numbers is enumerable.
  8. Show that a union of finite or countably many sets each of which is finite or countable is either finite or countable.
  9. Prove the set of transcedental numbers is not countable.
  10. Prove there exist infinite sets which are neither countable nor equivalent to the continuum. [Hint: conside all the real functions on the closed interval [0,1].]
  11. Show that sometimes e^b is irrational and sometimes it is rational as b ranges over the irrationals. [hint chap 6]
  12. A map f:X->Y induces a map f* from the power set of X to the power set of Y. Is every g from the power set of X to power set of Y a f* for some f:X->Y? Justify.
  13. If X is nonempty set, then card X < card Y iff there is an onto function g:Y -> X.
  14. card X ≤ card Y iff there is a 1-1 function g:X -> Y.
  15. Show 1-1 and onto are orthogonal properties. There are examples of all possible combinations of these properties.
  16. union of a finite or countable collection of sets each with card c has card c.
  17. the cartesian product of two sets with card c has card c.
• Chapter 17
  1. For Chebyshev polynomials T_n(x) and n > m, T_n(x) T_m(x) = [T_(n+m)(x) + T_(n-m)(x)]/2.
  2. Consider approximating a third degree polynomial p(x) by a quadratic q(x) so that max { |p(x) - q(x)| : -1 ≤ x ≤ 1} is as small as possible.
• Chapter 19
  1. Using the "Herglotz trick" prove that pi^2 csc^2 pi z = sum -infinity to infinity of (z - n)^-2.
  2. If f(x+y) = f(x)f(y) for a function f:|R -> |R+ and f(1)=a, then prove that f(x) = a^x. [Need f continuous.]
  3. Give asymptotically tight bounds on the following (z, s positiveintegers)
     1. sum i from 1 to n of i^z
     2. sum i from 1 to n of log^s i
     3. sum i from 1 to n of i^z log^s i
     4. sum i from 1 to floor(log(n)) of ceiling (n/2^i)
  4. show 2n+1 choose n ≥ 2^(2n) for n ≥ 0
  5. let S(n,z) = 1^z + 2^z + ... + n^z for positive integers n, z show S(n,z) = [1/(z+1)]((n+1)^(z+1) -[ (z+1 choose 2) S(n,z-1) + (z+1 choose 3) S(n, z-2) + .. (z+1 choose z) S(n,1) + (n+1) ]).
  6. Shikawa(1934) proved pi(xy) > pi(x) + pi(y) for x ≥ 6, y ≥ 2. Ues this result to prove Bertrand's Postulate. [pi(x) = number of primes ≤ x.]
• Chapter 21
  1. Let n > 2 and let a_1 ... a_n be integers, then two of the a_n are congruent mod (n-1).
  2. Consider the k*n numbers 1, 2, ... k*n and let A be a subset of (k-1)*n+1, show at least one number is divisible by k.
  3. Let G be a simple graph with n vertices. Any path of length n or greater contains a cycle. [Path is the wrong word]
  4. n(n-1)...(n-p+1)2^(n-p) = sum_k=p to k=n (n choose k)k(k-1).. (k-p+1) when p <= n.
  5. Let A be n + k elements from {1, 2, .. 2n}. Then there are k numbers in A that are divisible by some other number in A. [This isn't clear, so `guess' and put a proof or disprove in front.]
  6. Let S be a relation on X x Y given by y = x^2 where X={1,2,3} and Y={1 ... 9}. Compute the incident matrix for S.
  7. Show that on 5 vertices the maximal number of edges is 6 and that the butterfly |><| is the only graph on 5 vertices and 6 edges that has no 4 cycle (p. 136)
  8. Prove every graph with 2 or more vertices has at least two vertices with the same degree. (This is false for general graphs, but true for simple graphs.)
  9. Use the pigeonhole principle to prove that the decimal expansion of a rational number m/n eventually is repeating.
  10. Fifty-one points are scattered inside a square with a side of 1 meter. Prove that some set of three of these points can be covered by a square with side 20 cm.
  11. Show that an equilateral triangle cannot be completely covered by 2 smaller equilateral triangles http://www.math.uci.edu/~mathcirc/math194/lectures/pigeon/node3.html
  12. If a city has 10,000 different telephone lines numbered by 4-digit numbers and more than half of the telephone lines are in the downtown, then there are two telephone numbers in the downtown whose sum is again the number of a downtown telephone line.
  13. If there are 6 people at a party, then either 3 of them knew each other before the party or 3 of them were complete strangers before the party.
  14. Suppose a musical group has 11 weeks to prepare for opening night, and they intend to have at least one rehearsal each day. However, they decide not to schedule more than 12 rehearsals in any 7-day period, to keep from getting burned out. Prove that there exists a sequence of successive days during which the band has exactly 21 rehearsals. http://www.mathpages.com/home/kmath389.htm
• Chapter 22
  1. Prove that for n > 2k, the only maximal intersecting families are those which have a fixed element.
  2. Read about the inclusion-exclusion principle
  3. If Y is a subset of the finite set X, then Y is finite.
  4. If f is a function and X is finite, then f(X) is finite.
  5. If X is finite set then power set of X has 2^|X| elements.
• Chapter 24
  1. There are 125 label trees on {1, 2, 3, 4, 5} divide them into isomorphism classes.
  2. Draw simple graphs (if possible) in which the degree of the vertices are (a) 2, 2, 2, 2, 2; (b) 2, 3, 4, 5, 6; (c) 3, 3, 4, 5, 5, 6.
  3. A connected graph on n vertices has at least n-1 edges.
  4. If P is a path graph, show there is an unique path between any two vertices of P.
  5. If e is an edge in a connected graph G, show there is a spanning subtree of G which contains e.
  6. If T is tree subgraph of the complete graph G then all the edges of G-T are in one component.
  7. If the simple graph G has n vertices, n-1 edges and no circuits, then G is a tree.
  8. If G has exactly two vertices x and y of odd degree, then there is a path in G from x to y.
• Chapter 27
  1. Find an uncolored map of the US and color it with as few colors as possible. Proof that you have used the minimum number of colors needed.
  2. Good exercise: Find a 7-color map of the torus. Fun to try, and a nice reminder that coloring has something to do with topology. I got a very ugly answer when I tried. A prettier answer, along with more on the topic, may be found here: http://www.mimuw.edu.pl/delta/delta7/mapy/mapy.htm
• Chapter 28
  1. Triangulate the two figures on page 181
  2. Create another n-walled museum which requires floor(n/4) guards to guard it, given the variation suggested at the end of the section
• Chapter 29
  1. Define: k-clique, clique number
  2. Draw K_4
  3. State Turan's bound
  4. Explain the construction demonstrating that this bound is sharp.
  5. Express the sum of the degrees of the vertices of a graph as a function of the number of edges.
  6. Explain why we can bound |E| by maximizing f(w) = sum {w_i w_j : (v_i, v_j) edges } where w is a probability distribution on the vertices.
• Chapter 32
  1. If 2 ≤ s ≤ m and 2 ≤ t ≤ n, then R(s,t) ≤ R(m,n) with equility iff t=n and s=m.
  2. Prove R(3,n) ≤ (n^2+n)/2
  3. Let m, n ≤ 2 and p = R(m,n)-1. If the edges of G=K_p are colored red or blue then both (G contains red K_m-1 or blue K_n) and (G contains red K_m or blue K_n-1).
  4. In a room there are 10 people whose ages are integers between one and sixty. Show there are non overlapping subsets of people with the same sum of ages.
• Filters
  1. If F is a free ultrafilter, A is in F and B is finite then A-B in F.
  2. If F is filter on X and A is in F, then G = {B intersect A, B in F} is a filter on A. Furthermore, if F is an ultrafilter so is G.