Exercise 15.3. Suppose a and b are complex numbers. 1. Verify that N (ab) = N...
Theorem 16.1. Let p be a prime number. Suppose r is a Gaussian integer satisfying N(r) = p. Then r is irreducible in Z[i]. In particular, if a and b are integers such that a² +62 = p, then the Gaussian integers Ea – bi and £b£ai are irreducible. Exercise 16.1. Prove Theorem 16.1. (Hint: For the first part, suppose st is a factorization of r. You must show that this factorization is trivial. Apply the norm to obtain p=...
From last part of Q49 Exercise 1.49. The norm of a is the product of a and its complex conjugate: N(a) = aa. Ifa = x +yi, then N(a) is the square of the distance from (0,0) to (x.y) in the complex plane. If a and b are complex numbers, then N(ab)- N(a)N(b). If a is in G, the norm of a is a nonnegative member of Z. If a and b are in G and a divides b in...
Suppose A and B are matrices with matrix product AB. If bi, b2, ..., br are the columns of B, then Ab, Ab2, ..., Ab, are the columns of AB 1. Suppose A is an nxnmatrix such that A -SDS where D diag(di,d2,... dn) is a diagonal matrix, and S is an invertible matrix. Prove that the columns of S are eigenvectors of A with corresponding eigenvalues being the diagonal entries of D Before proving this, work through the following...
Letf: AB be a function and A1.A2 CAbe subsets of the domain. Show that fAinA2) fAANAA2) a. b. Can you find a condition on fx so that in this formula could be replaced byExplain. c. If m,n are integers and n is positive, prove the following identitty: d. Show that log(n!)-O(nlogn) e. An integerm e Z is called a composite number if m is divisible by some other integere d1. For an integer numbers 2 2, show that all of...
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers na +3n+6 and n2 + 2n +7 cannot be prime at the same time. (b) Find 15261527863698656776712345678%5 without using a calculator. (c) Let a be an integer number. Suppose a%2 = 1. Find all possible values of (4a +1)%6. 2. (Integers: %, =) (a) Suppose a, b, n are integer numbers and n > 0. Prove that (a+b)%n = (a%n +B%n)%n. (b) Let a,...
Let R denote the ring of Gaussian integers, i.e., the set of all complex numbers a + bi with a, b ∈ Z. Define N : R → Z by N(a + bi) = a^2 + b^2. (i) For x,y ∈ R, prove that N(xy) = N(x)N(y). (ii) Use part (i) to prove that 1, −1, i, −i are the only units in R.
Exercise 7 (2 points) Recall the binomial coefficient for integer parameters 0 Sk< n. Prove that Exercise 8 (2 points) Prove the following: if z is an integer with at most three decimal digits aia2a3, then x is divisible by 3 if and only if aut a2 +a3 is divisible by 3. Exercise 9 (3 points) A square number is an integer that is the square of another integer. Let x and y be two integers, each of which can...
a) Show that [a,b] | ab. b) Let d be a common divisor of a and b. Show that . c) Prove that (a,b)*[a,b] = ab. d) Prove that if c is a common multiple of a and b, then such that k[a,b] = c. e) Suppose that c is a common multiple of a and b. Show that ab | (a,b)*c Defn: Let m e Z. We say that m is a common multiple of a and b if...
1. Suppose a and b are elements of a group G. Prove, by induction, (bab−1)n = banb−1 . Hence prove that if a has order m, then bab−1 also has order m. Deduce from question (#1) that in any group ab and ba have the same order (you may assume ab has finite order). The assertion in Question (#1) can be generalized to an assertion about isomorphisms. State and prove it.
1. [10 marks] Modular Arithmetic. The Quotient-Remainder theorem states that given any integer n and a positive integer d there exist unique integers q and r such that n = dq + r and 0 r< d. We define the mod function as follows: (, r r>n = qd+r^0<r< d) Vn,d E Z d0 Z n mod d That is, n mod d is the remainder of n after division by d (a) Translate the following statement into predicate logic:...