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,
![{\displaystyle \alpha (F_{p_{1}^{e_{1}}}F_{p_{2}^{e_{2}}}\cdots \phi _{p_{k}^{2}})=\alpha ([F_{p_{1}^{e_{1}}},F_{p_{2}^{e_{2}}},\ldots ,\phi _{p_{k}^{2}}])=[p_{1}^{e_{1}},p_{2}^{e_{2}},\ldots ,p_{k}^{2}]=\phi _{\alpha (p)}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/d6be883dc2685ab010f9ff73f7dbbe45fb998119)
![{\displaystyle \alpha (F_{p_{1}^{e_{1}}}F_{p_{2}^{e_{2}}}\cdots \phi _{p_{k}^{2}}\cdot \phi _{p_{k}^{}})=\alpha ([F_{p_{1}^{e_{1}}},F_{p_{2}^{e_{2}}},\ldots ,\phi _{p_{k}^{2}},\phi _{p_{k}^{}}])=[p_{1}^{e_{1}},p_{2}^{e_{2}},\ldots ,p_{k}^{2},p_{k}^{}]=\phi _{\alpha (p)}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/cb97ac04f087f264848596f27722775da3661cc8)
However, we can measure that equality to verify that it is false.
![{\displaystyle \alpha (F_{p_{1}^{e_{1}}}F_{p_{2}^{e_{2}}}\cdots \phi _{p_{k}^{}})=\alpha ([F_{p_{1}^{e_{1}}},F_{p_{2}^{e_{2}}},\ldots ,\phi _{p_{k}^{}}])=[p_{1}^{e_{1}},p_{2}^{e_{2}},\ldots ,p_{k}^{}]=\phi _{\alpha (p)}/p}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/12151843b508c4e41e21b58fc1a56108581c8104)
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
.
.
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
,
.