 2.1: Each of the numbers1=1, 3 = 1+2, 6 = 1+2 + 3, 10 = 1+2 + 3 + 4, ......
 2.2: If tn denotes the nth triangular number, prove that in terms of the...
 2.3: Derive the following formula for the sum of triangular numbers, att...
 2.4: Prove that the square of any odd multiple of 3 is the difference of...
 2.5: In the sequence of triangular numbers, find the following:(a) Two t...
 2.6: (a) If the triangular number tn is a perfect square, prove that t4n...
 2.7: Show that the difference between the squares of two consecutive tri...
 2.8: Prove that the sum of the reciprocals of the first n triangular num...
 2.9: (a) Establish the identity tx = ty + tz, wheren(n + 3)x = + 12y=n +...
 2.10: Each of the numbers1, 5 = 1 + 4, 12 = 1 + 4 + 7, 22 = 1 + 4 + 7 + 1...
 2.11: For n 2:: 2, verify the following relations between the pentagonal,...
 2.12: Prove that, for a positive integer n and any integer a, gcd(a, a+ n...
 2.13: Given integers a and b, prove the following:(a) There exist integer...
 2.14: For any integer a, show the following:(a) gcd(2a + 1, 9a + 4) = 1.(...
 2.15: If a and b are integers, not both of which are zero, prove that gcd...
 2.16: Given an odd integer a, establish thata2 +(a+ 2)2 +(a+ 4)2 + 1is di...
 2.17: Prove that the expression (3n)!/(3!)n is an integer for all n =:::: 0.
 2.18: Prove: The product of any three consecutive integers is divisible b...
 2.19: Establish each of the assertions below:(a) If a is an arbitrary int...
 2.20: Confirm the following properties of the greatest common divisor:(a)...
 2.21: (a) Prove that if d I n, then 2d  1 12n 1.[Hint: Use the identity...
 2.22: Let tn denote the nth triangular number. For what values of n does ...
 2.23: If a I be, show that a I gcd(a, b) gcd(a, c)
Solutions for Chapter 2: EARLY NUMBER THEORY
Full solutions for Elementary Number Theory  7th Edition
ISBN: 9780073383149
Solutions for Chapter 2: EARLY NUMBER THEORY
Get Full SolutionsChapter 2: EARLY NUMBER THEORY includes 23 full stepbystep solutions. Since 23 problems in chapter 2: EARLY NUMBER THEORY have been answered, more than 5070 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Number Theory was written by and is associated to the ISBN: 9780073383149. This textbook survival guide was created for the textbook: Elementary Number Theory, edition: 7.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).