Question

Answer in FULL detail both proofs please!

Prove if 5 | Fn then 5 | n.

and then Prove

If 5 | n then 5 | Fn. Where Fn is the Fibonacci Numbers.

Answer #1

Please prove the following statement, in FULL detail. (by the if
and only if proving technique, not induction!)
Prove that 5 | Un if and only if 5|n. Where Un is the Fibonacci
sequence.

Please solve the following in FULL detail.
Using Un2 + (-1)n =
Un-1 * Un+1 to be true, Prove that any two
consecutive Fibonacci numbers are coprime.

Please answer in FULL detail!!
Prove that the Mobius function is NOT
Completely Multiplicative. Where completely
multiplicative is μ(m*n) = μ(m) * μ(n).

3. Prove the following about the Fibonacci numbers:
(a) Fn is even if and only if n is divisible by 3.
(b) Fn is divisible by 3 if and only if n is divisible by 4.
(c) Fn is divisible by 4 if and only if n is divisible by 6.

Please answer the following in FULL detail!
Is there a Pythagorean triple consisting of three Fibonacci
numbers? Give an example if there is one, or a proof if there
isn't.

Recall that the Fibonacci numbers are deﬁned by F0 = 0,F1 = 1
and Fn+2 = Fn+1 + Fn for all n ∈N∪{0}.
(1) Make and prove an (if and only if) conjecture about which
Fibonacci numbers are multiples of 3.
(2) Make a conjecture about which Fibonacci numbers are
multiples of 2020. (You do not need to prove your conjecture.)
How many base cases would a proof by induction of your
conjecture require?

Number Theory Course , I need a full explained answer for those
proofs please
1. Prove that for every integer x, x + 4 is odd if and only if x
+ 7 is even.
2. Prove that for every integer x, if x is odd then there exists an
integer y such that x^2 = 8y + 1.

The Fibonacci numbers are defined recursively as follows: f0 =
0, f1 = 1 and fn = fn−1 + fn−2 for all n ≥ 2.
Prove that for all non-negative integers n: fn*fn+2 = ((fn+1))^
2 − (−1)^n

Fibonacci Numbers.
The Fibonacci numbers are
1,1,2,3,5,8,13,21,….1,1,2,3,5,8,13,21,….
We can define them inductively by f1=1,f1=1, f2=1,f2=1, and
fn+2=fn+1+fnfn+2=fn+1+fn for n∈N.
Prove that fn=[(1+√5)n−(1−√5)n]/2n√5.

Prove that 5 | Un if and only if 5|n. Where Un is the Fibonacci
sequence.

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