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Learning Your ABC

Primes!

•Definition: A prime is a positive whole number that cannot
be divided evenly by anything except 1 and itself.

Examples

Awesome Property : Every rational number can be written
as a product of primes to a power .

Examples

Factorizations of Consectutive Numbers

Things We Notice

a and a + 1 have wildly different factorizations
As a gets really big, the factorizations become mostly "large"
primes ( like 2503) to a small power (like 1). (Demo)
Some rare gems are still "small" primes to a "large" power,
e.g.
We call these numbers smooth.
 

Relationships of Prime Factorizations

If I know the prime factorization of a and b, then it's easy to
find the prime factorization of ab.

Example

But what about a + b?

Example

ABC Triples

(smooth)*(smooth) = smooth
(smooth)+(smooth) = probably not smooth
  BUT SOMETIMES IT IS!

Examples

Judging ABC triples

Examples

Better Example: (0.96132)

Okay Example: (0.90013)

Best Example: (1.62991)

Typical Example: (0.36287)

Measuring ABC triples

Definition

The radical of a number , rad(n), is the product of all the primes
dividing n.

Example:

Definition

The ABC Ratio of a triple A + B = C is given by

Good ABC Triples

Top three known ABC ratio (verified up to 1020):

Definition

A good ABC triple is A + B = C where α(A, B, C) > 1.4.

•Largest known good ABC triples:

ABC Conjecture

ABC Conjecture (Oesterle and Masser, 1985)

For every η > 1, there exists only a finite number of ABC triples
such that

i.e. with

Consequences

Corollary

There is a largest α(A, B, C). It might be 1:62991.

Corollary

If the largest α < 2, then Fermat's Last Theorem (no integer
solutions to for n > 2) is proved.

Proof: Suppose there was a solution, then let A = xn, B = yn,
C = zn.
Then rad(ABC)≤xyz≤z3. Applying the conjecture gives
zn < (rad(ABC))2≤(z3)2 = z6. Hence n≤6.
The cases of 3≤n≤6 were proved in 1825 by Legendre and
Dirichlet.

More Consequences

Corollary

If the ABC conjecture is true then the following are also proved:

The generalized Fermat Mordell's conjecture
equation Roth's theorem
Wieferich primes statement Dressler's conjecture
The Erdos-Woods Bounds for the order of the
conjecture Tate- Shafarevich group
Hall's conjecture Vojta's height conjecture
The Erdos-Mollin-Walsh Greenberg's conjecture
conjecture The Schinzel-Tijdeman
Brocard's Problem conjecture
Szpiro's conjecture Lang's conjecture

... and many more!

How Close Are We to a Proof?

ABC Conjecture (Rephrased)

Given > 0, there exists a constant such that for every A, B, C
coprime integers with A + B = C,

where R = rad(ABC).

Theorem (Gyory (2007))

Let A, B, C be coprime integers with A + B = C. Let t be the
number of prime factors in R = rad(ABC). Then

An Analogy

Often a strong analogy between integers and polynomials with
rational coefficients .
A prime polynomial is one that cannot be factorized into
smaller polynomials with rational coefficents .
Example: x^2 + 1 is prime, but x^2 - 1 = (x + 1)(x - 1) is not.
(But x + 1 and x - 1 are.)
Let rad(P) be the product of all prime polynomials dividing P.
Example:
Let deg(P) be the degree of the polynomial. Notice that

deg(PQ) = deg(P) + deg(Q)

which is just like ln (AB) = ln(A) + ln(B).

The PQR Theorem

Replace A, B, C with polynomials P, Q, and R and replace ln
with deg.

PQR Theorem (Hurwitz, Stothers, Mason)

Let P, Q, R be nonconstant relatively-prime polynomials that
satisfy P + Q = R, then

deg(R) < deg(rad(PQR)):

PQR Proof

First notice that

Example:

F =
then F' =
so gcd(F, F') =
and =

What I Did

The ABC Conjecture can be generalized to number fields
Q() where is the root of a rational polynomial.

Example (Dokchitser)

This triple has algebraic ABC Ratio of 2.029.

There are "interesting" surfaces in algebraic geometry with
"special" points that correspond to algebraic numbers .
The corresponding algebraic numbers satisfy and
are usually smooth.
I used some algorithms developed in my thesis to generate 350
of these examples and computed their algebraic ABC ratios.

Results

•Points given in order of the degree of the defining polynomial.
•None of these "special" points correspond to a good ABC
example.
•Data does follow a trend. Proof? No idea how to even begin.
•Failure? Well, yes, but no.

The End?

Thanks!

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