Jump to content

User talk:Primedivine

Page contents not supported in other languages.
From Wikipedia, the free encyclopedia

Main idea

[edit]

Let .
For all divisors , where .
For proper divisors , where .



Greatest common divisor

[edit]

- Lucas
For , . - SF

For all divisors of , the antecedent can be swapped with the consequent.
If then .
If then .

Entry point of divisors

[edit]

All positive divisors divide some Fibonacci number.
Let denote the least positive Fibonacci number divisible by , such that .
Let denote the least positive Fibonacci number in the sub sequence that is divisible by , such that .
For , the entry point of a positive Fibonacci number is simply the subscript , ie .

Methods

[edit]

Primitive prime powers

[edit]

For , each Fibonacci number will have at least one primitive prime divisor, by Carmichael's theorem. By the Wall-Sun-Sun prime conjecture, could have at least one primitive prime power. Let denote the full product of primitive prime powers (one or more) that divide . By definition, this product of primitive prime powers always has an equal entry point to the whole Fibonacci number itself, ie .

Lowest common multiple

[edit]

For any positive integers a and b, let [a,b] denote the least common multiple of a and b.
- D. W. Robinson April 1963

If through are relatively prime then through are also relatively prime.


If through are relatively prime then we have the following.
Type A: or else
Type B: and also Twice the odd numbers, also called singly even numbers.
Type A:
Type B:

The fundamental theorem of arithmetic is bi-conditional with prime powered Fibonacci numbers. Let .
Type A: .
Type B:

  • .
  • .
  • .

Wall Sun Sun prime conjecture

[edit]

Let .
Suppose .
Suppose , for one or more Wall-Sun-Sun primes. In this particular instance, take for the sake of notation below.
Suppose and also , Type A.
If then .
If then , for , where .

Claim 1 (Right side b)

[edit]

If then .

Proof 1 (Right side b)

[edit]

. Solve for the products with the Robinson equality.



If , then , for divisors of .

Claim 2 (Left side a)

[edit]

If then .

Proof 2 (Left side a)

[edit]

If then .

.

Establish the hypothetical equality conjectured by Wall-Sun-Sun.
?

Solve for the products with the Robinson formula to prove that hypothetically a Wall Sun Sun prime would cause this equality to be true.

Claim 3 (Invalidate the conditional of Claim 2)

[edit]


Proof 3 (by contradiction)

[edit]

By the greatest common divisor, we have
.

By Wall's hypothesis,


By the Wall-Sun-Sun prime conjecture,







However, we can measure that equality to verify that it is false.

Proper divisors of the product of primitive prime powers

[edit]

By Carmichael's theorem, for will have at least one primitive prime divisor that has not appeared as a divisor of an earlier Fibonacci number. By the Wall-Sun-Sun prime conjecture, let denote the full product of primitive prime powers (one or more) that divide .

For proper divisors of ,
.

For ,
.

For example, if then
.
.




FTA

[edit]





Constructing Fibonacci numbers

[edit]

Let .
Let be proper divisors of n, composed of at least two distinct prime divisors.















Dirichlet

[edit]

, ie

Continued fractions for phi (golden ratio)

[edit]

It is well known that,
.
However,
.
Yielding,
,
,
,
, and so on.

Let .
yields
.

Let .
yields
.

Observe the related terms for and .
For all n, yields
,
.