- 9.1: Solve the following quadratic congruences:(a) x2 + 7x + 10 = 0 (mod...
- 9.2: Prove that the quadratic congruence 6x2 + 5x + 1 = 0 (mod p) has a ...
- 9.3: (a) For an odd prime p, prove that the quadratic residues of p are ...
- 9.4: Show that 3 is a quadratic residue of 23, but a nonresidue of 31
- 9.5: Given that a is a quadratic residue of the odd prime p, prove the f...
- 9.6: Let p be an odd prime and gcd(a, p) = 1. Establish that the quadrat...
- 9.7: If p = 2k + 1 is prime, verify that every quadratic nonresidue of p...
- 9.8: Assume that the integer r is a primitive root of the prime p, where...
- 9.9: (a) If ab= r (mod p), where r is a quadratic residue of the odd pri...
- 9.10: Let p be an odd prime and gcd( a, p) = gcd( b, p) = 1. Prove that e...
- 9.11: (a) Knowing that 2 is a primitive root of 19, find all the quadrati...
- 9.12: If n > 2 and gcd( a, n) = 1, then a is called a quadratic residue o...
- 9.13: Show that the result of the previous problem does not provide a suf...
- 9.14: (a) If the prime p > 3, show that p divides the sum of its quadrati...
- 9.15: Prove that for any prime p > 5 there exist integers 1 :S a, b :S p ...
- 9.16: (a) Let p be an odd prime and gcd(a, p) = gcd(k , p) = 1. Show that...
- 9.17: Prove that the odd prime divisors p of the integers 9n + 1 are of t...
- 9.18: For a prime p = 1 (mod 4), verify that the sum of the quadratic res...
- 9.19: Derive the Generalized Quadratic Reciprocity Law: If a and b are re...
- 9.20: Using the Generalized Quadratic Reciprocity Law, determine whether ...
Solutions for Chapter 9: EULER'S CRITERION
Full solutions for Elementary Number Theory | 7th Edition
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).
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].
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
A sequence of steps intended to approach the desired solution.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
A directed graph that has constants Cl, ... , Cm associated with the edges.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.