 11.1: For an even perfect number n > 6, show the following:(a) The sum of...
 11.2: Prove that the Mersenne number M19 is a prime; hence, the integer n...
 11.3: Prove that the Mersenne number M29 is composite.
 11.4: A positive integer n is said to be a deficient number if O' (n) < 2...
 11.5: Assuming that n is an even perfect number and d In, where 1 < d < n...
 11.6: Prove that any multiple of a perfect number is abundant.
 11.7: Show that no proper divisor of a perfect number can be perfect.[Hin...
 11.8: Find the last two digits of the perfect numbern = 219936(219937 _ l)
 11.9: If a(n) =kn, where k ::=::: 3, then the positive integer n is calle...
 11.10: Show that 120 and 672 are the only 3perfect numbers of the form n ...
 11.11: A positive integer n is multiplicatively perfect if n is equal to t...
 11.12: (a) If n > 6 is an even perfect number, prove that n = 4 (mod 6).[H...
 11.13: For any even perfect number n = 2k1(2k  1), show that 2k I a(n2) ...
 11.14: Numbers n such that cr(cr(n)) = 2n are called superpeifect numbers....
 11.15: The harmonic mean H (n) of the divisors of a positive integer n is ...
 11.16: The twin primes 5 and 7 are such that one half their sum is a perfe...
 11.17: Prove that if 2k  1 is prime, then the sum2kl + 2k + 2k+l + ... +...
 11.18: Assuming that n is an even perfect number, say n = 2kl (2k  1 ), ...
 11.19: If n1, n2, ... , nr are distinct even perfect numbers, establish th...
 11.20: Given an even perfect number n = 2k1(2k  1), show that(n) = n  2...
Solutions for Chapter 11: PERFECT NUMBERS
Full solutions for Elementary Number Theory  7th Edition
ISBN: 9780073383149
Solutions for Chapter 11: PERFECT NUMBERS
Get Full SolutionsChapter 11: PERFECT NUMBERS includes 20 full stepbystep solutions. Elementary Number Theory was written by and is associated to the ISBN: 9780073383149. Since 20 problems in chapter 11: PERFECT NUMBERS have been answered, more than 5585 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Number Theory, edition: 7.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Iterative method.
A sequence of steps intended to approach the desired solution.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).