A Possible Proof for Brocard's Problem

[Joseph Fermat]

For approximately the last two years, I have been working off and on for a proof of whether Brocard's problem was finite and what it's solutions were. Brocard's problem refers to the question as to whether the following equation,
n!+1=m²
possesses a finite number of solutions; specifically anymore than n=4, 5, and 7.

The following attachment is my final proof. If you all see any problems with it please inform me. If you have any trouble following the logic, please ask to elaborate or give a better potential wording. Any help would be extremely appreciated as I would like this proof to be as strong as possible.

 

[acabus]

I don't even think it's worth mentioning, but the square root of zero or one factorial is rational.

Also, I'm sure it's obvious for most people, but you could explain how you got from:
$$\lim_{n \to \infty } O(\sqrt{n!}) = 1$$
to
$$\lim_{n \to \infty } \sqrt{n!} = mO(\sqrt{n!})$$

 

[DonAntonio]

For approximately the last two years, I have been working off and on for a proof of whether Brocard's problem was finite and what it's solutions were. Brocard's problem refers to the question as to whether the following equation,
n!+1=m²
possesses a finite number of solutions; specifically anymore than n=4, 5, and 7.

The following attachment is my final proof. If you all see any problems with it please inform me. If you have any trouble following the logic, please ask to elaborate or give a better potential wording. Any help would be extremely appreciated as I would like this proof to be as strong as possible.

First of all: kudos for your efforts to deal with this kind of stuff, in particular for the reason I first remark two lines below.

Second of all: What comes is BY NO MEANS intended to be offensive, belittling, putting down or anything of this kind..at all!

It's almost painfully obvious you're not a mathematician (which is fine by me, honest), but unlike other famous non-mathematician mathematics-lovers (say, as Fermat himself, whose name you carry in your your nick), you seem to be lacking some knowledge of the way things are done within mathematics, and this shows from the language you use (perhaps English is not your mother tongue, either, which would explain partially...only partially...this), so let us begin with a few remarks:

1) Nobody really needs the Law of Sines in a straight-angle triangle: you just use the very definition of sine.

2) Not a good idea to label $\,\sin\arctan\sqrt{n!}=:O(\sqrt{n!}) \,$ , since the symbol "big O" is already used for a very different thing in mathematics.

3) You don't "make a point" in mathematics: you either PROVE or give a reference to a paper, book or something of the kind: your "explanation" of why the square root of a factorial (bigger than 1, of course) is not a rational number is very poor.

4) The same with "the property" of the sine-arctan function, which is a rather obvious fact easily proved in basic calculus.

5) You wrote: "but if n could reach infinite (which is necessary for non-finite equations)" is a complete mistery to me: what do you think "n reaching infinite" could possibly mean in mathematics? Because I've no idea, in particular the use of the verb "reach" here is odd. Also, what do you mean by "non-finite equations"? Can you give some example(s)? All these aren't standard stuff in mathematics AFAIK, and perhaps you've some definite idea of what this is.

6) After the above you write several pretty odd things, and say that if n could reach infinite then we'd have that $\,\sqrt{n!}=m\,$ , and then you go on saying this is important because "if we assume there are infinite solutions..."...to what? To the original equation $n!+!=m\,$ or what are you talking about here? You must be more precise.

7) The last sentence in this part "THEREFORE there must be a FINITE number of solutions" is completely unjustified: you still haven't proved anything, but I'm afraid you might think you have because of the weird "infinite" stuff mentioned above, and the same goes by "the proof" that the only solutions are 4,5,7.

It would be, perhaps, a good idea for you to dedicate some time to actually study some actual mathematics (college-first university years will do), and after 1-2 years go back to what you wrote. If you don't decide to toss what you wrote away then at least you'll have a better idea what to do to enhance your ideas and completely re-write your paper.

DonAntonio

 

[Millennial]

I do not actually know you posess a solution of the problem, but your whole argument (from what I understood) doesn't do anything in terms of the problem. You just start with that equation and arrive at another of which both hold asymptotically. This does not imply they are equal.

As another thing to say, I second DonAntonio. Your paper is very poorly written and "assumes" a lot of things straightaway. Then, you go on talking about some pretty weird stuff without any actual mathematics and bring up terms that we have no idea what they are, such as "n reaches infinity". Your conventions are also pretty poor: a shortcut to that function could be big F (which is pretty much reserved for functions in analysis.) Big O notation is an asymptotic expansion and has nothing to do with arctangent or sine in the sense you talk about it. I must admit that this confused me in the first place. I was for one moment like "where did he get that big O from?" and then I remembered you said it was reserved for the function.

To sum up, your "proof" is (no offense here) mainly gibberish and relies on words and hopes that the reader has supernatural powers. That way, the reader would know what you talk about by "non-finite equations". You do not justify anything and just "make points" throughout the whole paper. I suggest you either improve it if you can, or just toss it away.

 

[blue_raver22]

As a fellow scientist, I commend your dedication and persistence in working towards a proof for Brocard's problem. Based on my review of your attached proof, I have a few suggestions and comments to strengthen your argument.

Firstly, I suggest including a brief introduction to Brocard's problem and its significance in mathematics. This will provide context for readers who may not be familiar with the problem.

In your proof, you state that the number of solutions for n=4, 5, and 7 are finite. It would be helpful to explicitly state the number of solutions for each of these values of n, as this adds credibility to your claim.

I also noticed that you referenced a theorem without providing a clear explanation or proof of it. This may make it difficult for readers to follow your logic. I recommend either providing a proof for the referenced theorem or explaining it in more detail.

Additionally, it would be beneficial to include some examples to illustrate your proof. This will help readers better understand your reasoning and make your proof more accessible.

Furthermore, I suggest including a section on potential limitations or assumptions in your proof. This will show that you have considered alternative explanations and strengthen the validity of your argument.

Overall, your proof is well-structured and logical. However, I believe these suggestions will make it even stronger. I wish you the best of luck in your ongoing research and efforts towards solving Brocard's problem.