Let X Geom(p). For positive integers n, k define
P(X = n + k | X > n) = P(X = n + k) / P(X > n) :
Show that P(X = n + k | X > n) = P(X = k) and then briefly
argue, in words, why this is true for geometric random
variables.
Let p be an odd prime (i.e., any prime other than 2). Form two
vector spaces V1, V2 over Fp
(prime field of order p) with bases corresponding to the edges and
faces of an icosahedron (so that V1 has dimension 30 and
V2 has dimension 20). Let
T : V1 → V2 be the linear transformation
defined as follows: given a vector v ∈ V1, T(v) is the
vector in V2 whose component corresponding to a given
face is...
Let p be an odd prime.
(a) (*) Prove that there is a primitive root
modulo p2 . (Hint: Use that if a, b have orders n, m,
with gcd(n, m) = 1, then ab has order nm.)
(b) Prove that for any n, there is a primitive
root modulo pn.
(c) Explicitly find a primitive root modulo
125.
Please do all parts.
Thank you in advance
Let p be an odd prime and a be any integer which is not
congruent to 0 modulo p.
Prove that the congruence x 2 ≡ −a 2 (mod p) has solutions if
and only if p ≡ 1 (mod 4).
Hint: Naturally, you may build your proof on the fact that the
statement to be proved is valid for the case a = 1.
Let A be a diagonalizable n × n matrix and let P be an
invertible n × n matrix such that B = P−1AP is the diagonal form of
A. Prove that Ak = PBkP−1, where k is a positive integer. Use the
result above to find the indicated power of A. A = 6 0 −4 7 −1 −4 6
0 −4 , A5 A5 =
7. Let n ∈ N with n > 1 and let P be the set of polynomials
with coefficients in R.
(a) We define a relation, T, on P as follows: Let f, g ∈ P. Then
we say f T g if f −g = c for some c ∈ R. Show that T is an
equivalence relation on P.
(b) Let R be the set of equivalence classes of P and let F : R →
P be...
Let A be a diagonalizable n × n
matrix and let P be an invertible n × n
matrix such that
B = P−1AP
is the diagonal form of A. Prove that
Ak = PBkP−1,
where k is a positive integer.
Use the result above to find A5
A =
4
0
−4
5
−1
−4
6
0
−6
1)Show that a subset of a countable set is also countable.
2) Let P(n) be the statement that 13 + 23 +· · ·+n3 =(n(n +
1)/2)2 for the positive integer n.
a) What is the statement P(1)?
b) Show that P(1) is true, completing the basis step of
the proof.
c) What is the inductive hypothesis?
d) What do you need to prove in the inductive step?
e) Complete the inductive step, identifying where you
use the inductive hypothesis....
Number Theory:
Let p be an odd number. Recall that a primitive root, mod p, is
an integer g such that gp-1 = 1 mod p, and no smaller
power of g is congruent to 1 mod p. Some results in this chapter
can be proved via the existence of a primitive root(Theorem
6.26)
(c) Given a primitive root g, and an integer a such that a is
not congruent to 0 mod p, prove that a is a square...