•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,
•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.
•But what about a + b?
•(smooth)*(smooth) = smooth •(smooth)+(smooth) = probably not smooth
BUT SOMETIMES IT IS!
A good ABC triple is A + B = C where α(A, B, C) >
•Largest known good ABC triples:
ABC Conjecture (Oesterle and Masser, 1985)
For every η > 1, there exists only a finite number of
There is a largest α(A, B, C). It might be 1:62991.
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
If the ABC conjecture is true then the following are also
Replace A, B, C with polynomials P, Q, and R and replace
PQR Theorem (Hurwitz, Stothers, Mason)
Let P, Q, R be nonconstant relatively-prime polynomials
satisfy P + Q = R, then
deg(R) < deg(rad(PQR)):
•First notice that
so gcd(F, F')
What I Did
The ABC Conjecture can be generalized to number fields
is the root of a rational polynomial.
•This triple has algebraic ABC Ratio of 2.029.
There are "interesting" surfaces in algebraic geometry
"special" points that correspond to algebraic numbers . •The corresponding algebraic numbers satisfy
are usually smooth. •I used some algorithms developed in my thesis to generate 350
of these examples and computed their algebraic ABC ratios.
•Points given in order of the degree of the defining
•None of these "special" points correspond to a good ABC
•Data does follow a trend. Proof? No idea how to even begin.
•Failure? Well, yes, but no.
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