Congruences, Primitive Roots, Indices and Table of Indices

6. Let g be a primitive root of m. An index of a number a to the base (written ing a) is a number + such that g+≡a(mod m). Given that g is a primitive root modulo m, prove the following...

7. Construct a table of indices of all integers from....

8. Solve the congruence 9x≡11(mod 17) using the table in 7.

9. Suppose g is a primitive root modulo P (a prime) and m|p-1 (1<m<p-1). How many integral solutions are there of the congruence x^m - g ≡ (mod p) ?

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** Please see the attached file for the complete problem description **
I have a table showing Holt's seasonal forecast with multiplicative seasonal indices. In this model, beta is twice the value of alpha, the seasonal indices average to 1 and the seasonal factors for quarter 1 and quarter 3 are both equal to 1.
Please c

Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitiveroots, quadratic reciprocity and quadratic fields.
(See attached file for full problem description)

Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitiveroots, quadratic reciprocity and quadratic fields.
(See attached file for full problem description)

1. If g is a primitive root of p, show that two consecutive powers of g have consecutive least residues. That is, show that there exists k such that g^(k+1)=g^k+1(mod p) (Fibonacci primitive root)
2. Show that if p=12k+1 for somek , then (3/p)=1
3. Show that if a is aquadratic residue (mod p) and ab=1(mod p) then b is a qu

I need help with this problem. It has two parts a) and b).
(a) Show that if m is a number having primitiveroots, then the product of the positive integers less than or equal to m and relatively prime to is congruent to -1(mod m).
(b) Show that the result in (a) is not always true if m does not have primitive roots.

Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitiveroots, quadratic reciprocity and quadratic fields.
(See attached file for full problem description)

(See attached file for full problem description with all symbols)
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Suppose that n is odd and a is a primitive root modulo n.
(a) Show that there exists and integer b such that and .
(b) Show that b is a primitive root modulo 2n.

Raise the quantity in parentheses to the indicated exponent, and simplify the resulting expression. Express answers with positive exponents.
1. ( x^-2 y^-1 / 2x^0 y^3 )^3
2. ( 24y^-1 / 72x^-3 y^4)^2

Show that the general direction [ hkl ] in a cubic crystal is normal to the planes with Miller indices (hkl). Is the same true in general for an orthorhombic crystal?
Show that the spacing d of the (hkl) set of planes in a cubic crystal with lattice parameter a is:
d = (a)/(h^2 + k^2 +l^2)^(1/2)
What is the generaliza