posted on Sep, 12 2012 @ 11:54 PM
reply to post by syrinx2112
Ahh you got me thinking here... Good thought on this. Please elaborate on the "prediction" part (yes you did state potentially)... I am
curious and want to make sure I am following you..
It's perhaps my lack of understanding regarding mathematical theories. I tend to be more object oriented in my thinking.
From what I'm gathering on a brief wikipedia primer (dangerous to draw too many conclusions off of that, but it helps to kick start one's keyword
The abc conjecture has a large number of consequences. These include both known results, and conjectures for which it gives a conditional
Thue–Siegel–Roth theorem on diophantine approximation of algebraic numbers
Fermat's Last Theorem for all sufficiently large exponents (proven in general by Andrew Wiles)
The Mordell conjecture (Elkies 1991)
The Erdős–Woods conjecture except for a finite number of counterexamples (Langevin 1993)
The existence of infinitely many non-Wieferich primes (Silverman 1988)
The weak form of Marshall Hall's conjecture on the separation between squares and cubes of integers (Nitaj 1996)
The Fermat–Catalan conjecture, a generalization of Fermat's last theorem concerning powers that are sums of powers (Pomerance 2008)
The L function L(s,(−d/.)) formed with the Legendre symbol, has no Siegel zero (this consequence actually requires a uniform version of the abc
conjecture in number fields, not only the abc conjecture as formulated above for rational integers)
P(x) has only finitely many perfect powers for integral x for P a polynomial with at least three simple zeros.
A generalization of Tijdeman's theorem
It is equivalent to the Granville–Langevin conjecture.
It is equivalent to the modified Szpiro conjecture, which would yield a bound of \operatorname[rad](abc)^[\frac+\epsilon] (Oesterlé
Dąbrowski (1996) has shown that the abc conjecture implies that n! + A= k2 has only finitely many solutions for any given integer
While the first group of these have now been proven, the abc conjecture itself remains of interest, because of its numerous links with deep questions
in number theory.
This site goes into more detail:
Since this conjecture, again, if I'm understanding correctly, effectively establishes a constant between numbers with the greatest common factor of
one (or at least a sort of range), it could lead to far more efficient searches for prime numbers, if not a manner in which to predict a range within
which one could find prime numbers.
Though, again, my understanding of theoretical math is not all that stellar.
The implications for computer sciences, however, could be quite impressive if the proof holds.
This is not the first time a proof has been presented for the ABC conjecture, however:
Back in May of 2007, another individual proposed a proof that was later determined to not hold as proof.