Beal's conjecture prize

Originally Posted by MattcAnderson

Hi again everybody,

One of them was (sort of) recently solved.
Now, let's look at the Beal conjecture. It is a generalization of
Fermat's last theorem.

The Beal Conjecture is something like
a^x + b^y = c^z
where all variables are both positive and integers.

This is the sort of problem that definitely takes collaboration. One person cannot (yet) do it alone.
IMHO (In My Humble Opinion)

Matt, one of my tenets in reading through this forum is to recognize the source where conclusions can be drawn from that alone. Your post brought up (double entendre here) some points which you could address..such as the following:
Mathematical precision in statement, description and resolution must be considered. Wishy-washiness, inaccurate paraphrasing and bold hyperbole (closely affiliated with unfounded conjectures) should be avoided if you're aware of them.
I've been guilty of all of the above and have adapted (most of) my posts accordingly within this forum..one example of mine was mis-stating attributes of the critical line and I was brought to task on this. Chronic repetition of mistakes be they thematic or expository is not a desirable trend to maintain.

Regarding the first two points mentioned above, do your homework and learn more about both. If you read about the resolution of the Poincare Conjecture you will learn something about human nature which goes to your third point. Try to understand, to the best of your ability, this conjecture and its solution. Next, read about the "attribution" of the solution and what took place..and where the money eventually went.
Since I also like to play poker, (Yardley did as well..an interesting bloke), read "The Professor, the Banker, and the Suicide King." This conjecture is deep and I don't understand it as well as the one above. I have followed up with purported proofs which had some interesting approaches and have cross referenced them with reviewers to learn more. A good exercise to test your technical sagacity is to "spot the mistake in the proof" ...or any disqualified proof to address your mathematical chops..without peeking of course.

To your third point, read up on https://en.wikipedia.org/wiki/Grigori_Perelman rather than just being "barely familiar" with who he is. Conceptual advances made by committees are vastly overshadowed by those made by indivuals. I read the following novella years ago when it first came out and I recommend this updated version to you now..https://www.amazon.ca/SYSTEMANTICS-S.../dp/B00AK1BIDM

Regarding some of the other posts within this thread and references to pythagorean triangles, does anyone have a pointer to a comprehensive listing of congruent numbers (other than primes 5,6,7 mod 8)? Sardonicus?..anyone ..anyone?*

*If anyone watched "Win Ben Stein's Money" they'll get the reference. By the way, the show was hosted by Jimmy Kimmel and Ben Stein was an advisor to Nixon..both smart and funny men.
https://www.youtube.com/watch?v=zROOIwTlfgE


For the sake of completion, here's a link to Yardley's poker book, https://www.amazon.ca/Education-Poke.../dp/487187639X. He has written more books which are also "educational." However, due diligence is always required..https://ia600903.us.archive.org/21/i..._watermark.pdf.
A slight tangent will take you to this Canadian..https://en.wikipedia.org/wiki/William_Stephenson..a small plug for Canucks.

[xilman]

Originally Posted by jwaltos

Regarding some of the other posts within this thread and references to pythagorean triangles, does anyone have a pointer to a comprehensive listing of congruent numbers (other than primes 5,6,7 mod 8)? Sardonicus?..anyone ..anyone?*

*If anyone watched "Win Ben Stein's Money" they'll get the reference. By the way, the show was hosted by Jimmy Kimmel and Ben Stein was an advisor to Nixon..both smart and funny men.
https://www.youtube.com/watch?v=zROOIwTlfgE


For the sake of completion, here's a link to Yardley's poker book, https://www.amazon.ca/Education-Poke.../dp/487187639X. He has written more books which are also "educational." However, due diligence is always required..https://ia600903.us.archive.org/21/i..._watermark.pdf.
A slight tangent will take you to this Canadian..https://en.wikipedia.org/wiki/William_Stephenson..a small plug for Canucks.

That is one thing I really like about this place - I often learn something new and unexpected.

I had never heard of "Yardley's poker book" but own a 1st-edition copy of The American Black Chamber. I wondered if it was the same Yardley and followed your second link. Sure enough, it is. It would have been obvious if the full text of your third link had been visible but the forum had chopped out the interesting central portion.

[storm5510]

I did some coding experiments and came up with these, among many others:

(52572 ^ 21150) + (503141 ^ 24504) - (586043 ^ 196) <> 0
(411808 ^ 24036) + (673730 ^ 12731) - (321725 ^ 7257) <> 0
(18836 ^ 5853) + (3706 ^ 19816) - (30804 ^ 32334) <> 0

I read the pertinent part of this first. An excerpt:

a^x + b^y = c^z

a, b, c, x, y, and z are positive integers with (x, y, and z) > 2.
a, b, and c have a common prime factor.

Following kriesel's suggestion, I calculated for zero. I kept a scant few results to see how large they were. One was over 176,000 digits in length.

The OP wrote one person cannot do this alone, yet. As it is now, a great many, and their descendants, could run this for decades and never find a zero result. For me, it was simply a learning experience.

[storm5510]

FWIW: I still occasionally look at things on the web related to Beal. I found this. It might be useful to someone down the road.

[rebirther]

Originally Posted by storm5510

FWIW: I still occasionally look at things on the web related to Beal. I found this. It might be useful to someone down the road.

We had 2 BOINC projects in the past where I saved the source code but cant find it anymore, related to this I have found a pdf file link

[storm5510]

Originally Posted by rebirther

We had 2 BOINC projects in the past where I saved the source code but cant find it anymore, related to this I have found a pdf file link

Sorry, that is not a language I can read. Still, thanks for the reply.

[rebirther]

Originally Posted by storm5510

Sorry, that is not a language I can read. Still, thanks for the reply.

Tried to load in cache with google translator link

Otherwise copy / paste from pdf and translate

[storm5510]

Originally Posted by rebirther

Tried to load in cache with google translator link

Otherwise copy / paste from pdf and translate

Much better. Pairwise relatively prime?

The contents of the link I posted above can be confusing. The upper part says A, B, and C, must have a common prime factor. The lower says they cannot. It appears this is actually two different problems.

In any case, somebody or a group will come up with a resolution some day.

[LaurV]

How far did they test it? (I mean, the Polish people, Boinc, etc. - I didn't read the Polish paper).

I flirted for a while with Beal myself in the past, well... the million bucks is tempting, but this is a harder nut to crack compared with FLT itself... (of course, if easy, it would be already solved, and no money given ). The only lucky chance would be to find a counterexample, but as the "thing" is most probably true, you can grow quite old looking for one...

If you want just to "get famous" by verifying it to some higher limits, then a couple of imbricated (why is this underlined in red?) for-loops is NOT the solution. There is a lot of theory behind of how those triplets of numbers can look, and you would have to study that first, and impose a lot of conditions for how you chose the numbers. Otherwise, testing all possible combinations, the progress is extremely EXTREMELY slow. For example, you may want to avoid from the start pairs (a, b) with common factors. In that case, your program comes out empty-handed, and to make sure it does what it is supposed to do (and possibly show to other people later that you tested these ranges for real, not only claiming you did it), you should at least print something, for example, the "near" cases (there are many cases where the equality is "almost" true, differing by a very small number, 1, 2, or a very small epsilon, in case you check if a^x+b^y is a power of an integer.

If getting the money is your goal, and because finding a theoretical proof (or disproof) is very hard, and because the actual testing limits are quite impossible to reach (because people who know, used those theoretical knowledge, so you -general you -as a newcomer, don't have much of a chance to keep with them, but I also assume teams and individual people are hammering this hard, higher and higher, just in this moment, but they don't say the limits they reached, because they all want the money, and therefore don't want other teams and people "use" their limits, and search only further - which indeed is quite counter-productive, but well, people are selfish), then, your only "chance" is to search very high, random ranges, of such numbers. At least, pick one very large, random, to be sure nobody tested it in the past, and accommodate the others around it. Otherwise, you have huge chances to duplicate efforts done by other teams/people, who came out empty handed, but didn't tell you...

So, you may get lucky... But again, you may grow older trying...
Better use your computing resources for gimps
(and let me get the money... hihi... kidding... I am not working for Beal... or... if I would, would I tell you? )

[rebirther]

Originally Posted by LaurV

How far did they test it? (I mean, the Polish people, Boinc, etc. - I didn't read the Polish paper).

From the translation:

The first 5,000 compartments were tested using
ICM UW 3 supercomputing resources (it took in
approximately 600,000 computing hours 4 ). The last ones
601 intervals (9,999 <and <10,601), the
using both the BOINC platform and connected
to it supercomputer resources 5 using specially
configured environment. This gives approximately
additional 21,636 computing hours.