## 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

## 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,

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

## 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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