The Fermat numbers are numbers of the form 2 ^2n + 1 = Φn . Prove that if n < m , then Φn │ϕ m - 2.

The Fermat numbers are numbers of the form 2 ^2n + 1 = (Phi)n . Prove that if n < m , then (Phi)n │(Phi)m - 2.

Solution Summary

This solution is comprised of a detailed explanation for solving the problems on Fermat numbers.
It contains step-by-step explanation for proving statement that if n < m , then Φn │Φm - 2,
where Φn is the Fermat numbers such that Φn = 2 ^2n + 1.
It contains step-by-step explanation for proving statement that if n < m , then (Phi)n │(Phi)m - 2,
where (Phi)n is the Fermat numbers such that (Phi)n = 2 ^2n + 1.

A. show that if a < b
then a < 1/2(a+b) < b
b. i. if 0 < b < 1
show that
0 < b^2 < b < 1
ii. if 1 < b
show that
1 < b < b^2
c. show that abs(x-a) < E if an only if
a-E < x < a+E
where E= epsilon

Please help me learn how to write these two proofs correctly for my Modern Algebra class.
Please submit all work as either a PDF or MS Word file.
** Please see the attached file for the complete problem description **

Q12: (i) Calculate phi(15) in THREE ways.
(ii) Express in modular arithmetic
[hint:the number of integers from 1 to m that are relatively prime to m is denoted by phi(m). it is the number of elements in the set a:1=a=m and gcd(a,m)=1 ]

Proofs (see attached)
Question 1
Let b and d be distinct nonzero real numbers and c any real number .Prove that
{ b,c +di } is a basis of C over R.
Hint-For any r + si ∈ C, r +si= (r/b-cs/bd)b +s/d(c+di).Hence {b,c + di} spans C over R. Prove that it is also linearly independent over R.
Question 2
If a+bi ∈

First part is to find an expression in terms of n, the results of the formula:
and prove the expression is correct?
Secondly
A Fibonacci sequence is the basis for a superfast calculation trick as follows:
Turn your back and ask someone to write down any two positive integers (vertically and one below the othe

a. Let R^+ be the set of positive real numbers. Define operations on this set by the a + b = ab and a x b = e^(ln (a)ln (b)), where the right hand sides have the usual meaning in the real numbers. Prove that R^+ is a field with these operations by showing that exp : R -----> R^+ is an isomorphism, where exp is the usual expone

Let R(t) be the field of rational functions. Answer the following (with proofs):
A. If we identify the rational numbers with a subset of R(t), what function corresponds to a given rational number r?
B. Find a function f>0 in R(t) for which no rational number r satisfies 0 < r < f.