- 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 PHI-FUNCTION
- 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
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.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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 S-I AS = A = eigenvalue matrix.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Invert A by row operations on [A I] to reach [I A-I].
A symmetric matrix with eigenvalues of both signs (+ and - ).
A sequence of steps intended to approach the desired solution.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
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).
A directed graph that has constants Cl, ... , Cm associated with the edges.
Every v in V is orthogonal to every w in W.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.