 Chapter 1: Mathematical Induction
 Chapter 10: FROM CAESAR CIPHER TO PUBLIC KEY CRYPTOGRAPHY
 Chapter 11: PERFECT NUMBERS
 Chapter 12: THE EQUATION x2 + y2 = z2
 Chapter 13: SUMS OF TWO SQUARES
 Chapter 14: THE FIBONACCI SEQUENCE
 Chapter 15: FINITE CONTINUED FRACTIONS
 Chapter 16: PRIMALITY TESTING AND FACTORIZATION
 Chapter 2: EARLY NUMBER THEORY
 Chapter 3: THE FUNDAMENTAL THEOREM OF ARITHMETIC
 Chapter 4: BASIC PROPERTIES OF CONGRUENCE
 Chapter 5: FERMAT'S LITTLE THEOREM AND PSEUDOPRIMES
 Chapter 6: THE SUM AND NUMBER OF DIVISORS
 Chapter 7: EULER'S PHIFUNCTION
 Chapter 8: THE ORDER OF AN INTEGER MODULO n
 Chapter 9: EULER'S CRITERION
Elementary Number Theory 7th Edition  Solutions by Chapter
Full solutions for Elementary Number Theory  7th Edition
ISBN: 9780073383149
Elementary Number Theory  7th Edition  Solutions by Chapter
Get Full SolutionsThe full stepbystep solution to problem in Elementary Number Theory were answered by , our top Math solution expert on 03/14/18, 05:19PM. Elementary Number Theory was written by and is associated to the ISBN: 9780073383149. Since problems from 16 chapters in Elementary Number Theory have been answered, more than 4366 students have viewed full stepbystep answer. This textbook survival guide was created for the textbook: Elementary Number Theory, edition: 7. This expansive textbook survival guide covers the following chapters: 16.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

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.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.