Latest abc conjecture proof news

[only_human]

(nature.com) Proof claimed for deep connection between primes

If it is true, a solution to the abc conjecture about whole numbers would be an ‘astounding’ achievement.

This link does a Google search selecting the last 24 hours: "ABC conjecture" proof.

blogs:
On, Not Even Wrong. Proof of the abc Conjecture?:

Jordan Ellenberg at Quomodocumque reports here on a potential breakthrough in number theory, a claimed proof of the abc conjecture by Shin Mochizuki. More than five years ago I wrote a posting with the same title, reporting on a talk by Lucien Szpiro claiming a proof of this conjecture (the proof soon was found to have a flaw). One change over the last five years is that now there are excellent Wikipedia articles about mathematically important questions like this conjecture, so you should consult the Wikipedia article for more details on the mathematics of the conjecture. To get some idea of the significance of this, that article quotes my colleague and next-door office neighbor Dorian Goldfeld describing the conjecture as “the most important unsolved problem in Diophantine analysis”, i.e. for a very significant part of number theory.

The first link in that quote is for Quomondocumque, Mochizuki on ABC, where there is high level commentary.

mathoverflow: Philosophy behind Mochizuki’s work on the ABC conjecture

[LaurV]

This could turn out to be wonderful news! It may be the most important proof of the 21st century. No kidding. It worth a separate topic here.

Now my question is what should be the effect of this for Beal's Conjecture, and the bonus of 200k bucks.

Certainly if the ABC conjecture is true (and most probably it is, regardless of the fact if the current proof turns out to be correct or to have flaws), then there are only a finite number of solutions for Beal's problem. The award however is given for a specific solution (i.e. counterexample), or a proof that NONE exists. Proving that there are not many and that there is none is not the same thing.

Anyhow, certainly our don blazys totally lost his chance to prove it and take the money and the glory...
[edit: I am a bit nostalgic to this subject, I played with Beal's conjecture myself for a while, without any result, and this is how I came to mersenneforum.org, one of my friends pointed me to "the best source of fun i read in the last time" (his words, unfortunately he never posts here, but occasionally reading only, he says posting on forums is wasting of time), and there was my first post on this forum].

[only_human]

Originally Posted by LaurV

Now my question is what should be the effect of this for Beal's Conjecture, and the bonus of 200k bucks.

Dunno. Wikipedia has this list of consequences: abc conjecture#Some_consequences

The abc conjecture has a large number of consequences. These include both known results, and conjectures for which it gives a conditional proof.

• 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.^{[[URL="http://en.wikipedia.org/wiki/Abc_conjecture#cite_note-Ref_a-4"]5[/URL]]}
• A generalization of Tijdeman's theorem
• It is equivalent to the Granville–Langevin conjecture
• It is equivalent to the modified Szpiro conjecture.
• Dąbrowski (1996) has shown that the abc conjecture implies that n! + A= k² has only finitely many solutions for any given integer A.

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.

[rogue]

How about this:

Unified Theory or Matter

[xilman]

Originally Posted by rogue

How about this:

Unified Theory or Matter

Hmm. Interesting. I need to find out how this differs from previous attempts to add a scalar field to Einstein's field equations. AFAIK, previous attempts have either failed to be consistent with observations or have had only small and thus-far undetectable consequences, an example of the latter being the Brans-Dicke theory.

[xilman]

Originally Posted by xilman

Hmm. Interesting. I need to find out how this differs from previous attempts to add a scalar field to Einstein's field equations. AFAIK, previous attempts have either failed to be consistent with observations or have had only small and thus-far undetectable consequences, an example of the latter being the Brans-Dicke theory.

The paper looks fairly straightforward. Nicely written, anyway, though I don't follow all of it in any detail.

The basic assumption is that of the existence of scalar field \varphi which couples to the curvature to produce a modified gravitational field. \varphi is assumed to have zero divergence over the universe so, if not constant, must have both repulsive and attractive components, corresponding to dark energy and dark matter respectively. What I don't yet understand is why the two components aren't of equal magnitude. Observation suggests that dark energy is greater by a factor of several.

A nice point is that solutions to the modified field equations exist with T₀₀ non-zero. This, the time-time component of the stress-energy tensor, is identified with the energy density of the CMB.

Not yet seen anything grossly wrong with the theory, though I wouldn't expect to, and look forward to more sensitive tests of it.

[rogue]

Another claim for proof of Goldbach's Conjecture

I have not read the article yet, but the mathematician behind it does not appear to be a crank.