#HELP NUMBER THEORY PROBLEM.

171 messages · Page 1 of 1 (latest)

vapid fractal
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hello chat, i have something important to discuss about. is there anyone here in university study mathematics by any chance?
or studying maths beyond univeristy
please reach out to me.

Screenshot_2026-03-02_230059.png
torn parrotBOT
wispy panther
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I would imagine yes. If we consider the factorial to be a product of values of i, as in $\prod_{i=1}^n i = n!$, then each 3rd i will contribute one factor of 3, each 9th will contribute an additional factor and so on. Similarly with 5 and 7. 2n choose n is $(2n)!/(n!)^2$ so we only need to find an n such that it is only slightly larger than a power of 3, 5, and 7, so that no extra powers of these factors are contributed from a new prime power.

However, proving that you can always find such a number might be tricky. One way might be to prove that you can generate one of these numbers from one that already exists, but the action of multiplying by (for instance) 105 would compound differences between the number in question and each of the prime powers. Try playing around with this idea though.

sharp ploverBOT
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OmnipotentEntity

173636598150332416.png
vapid fractal
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thank you so much

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for your reply

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but

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can i have further help

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@wispy panther

wispy panther
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Well, I don't know the answer, and I suspect it might be an open problem

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Yes, it is open. This paper may be of interest https://arxiv.org/abs/2201.11274

arXiv.org
On a conjecture of Graham on the p-divisibility of central binomial...

We show that for every $r \geq 1$, and all $r$ distinct (sufficiently large) primes $p_1,..., p_r > p_0(r)$, there exist infinitely many integers $n$ such that ${2n \choose n}$ is divisible by these primes to only low multiplicity. From a theorem of Kummer, an upper bound for the number of times that a prime $p_j$ can divide ${2n \choose n}$ is ...

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@vapid fractal

vapid fractal
wispy panther
bleak dew
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anything to do with prime numbers bro

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I mean

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as long as n * n+1 *.....2n

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doesn't have a multiple of 5, 11, 105

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but then

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idk

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it seems more and more unlikely as n gets bigger

wispy panther
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More precisely. Let $n = \sum_i a_i \cdot 3^i$ be the expansion of $n$ in base 3. Similarly $n = \sum_i b_i 5^i$ and $n = \sum_i c_i 7^i$.

We want to avoid introducing an extra prime power when we double. This means we want to avoid a carry. In other words, for an $n$ to be admissible we require $a_i \in {0, 1}, b_i \in {0, 1, 2}, c_i \in {0, 1, 2, 3} \forall i$

In other words, we can use this criteria to hunt down these values. I just manually started with 50 (i.e. one of the first non-trivial values that might not be coprime to 105) and just kept heading up using the base expansions above until I found an example: $n = 756 = 2130_7 = 11011_5 = 1001000_3$

As you observe, these examples are more and more unlikely as n gets large because for a $k$-digit integer in a particular base selected uniformly at random each digit has $2/3$ chance of being admissible in base 3, a $3/5$ chance in base 5, and a $4/7$ chance in base 7 (these probabilities are not actually independent though, and, of course, the number has differing counts of digits in different bases).

sharp ploverBOT
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OmnipotentEntity

173636598150332416.png
wispy panther
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,w gcd((2*756) choose 756, 105)

sharp ploverBOT
wispy panther
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(however, I still think it is likely that there are infinitely many examples, because although the proportion of numbers that are admissible for a given base go to zero, as k gets large, the absolute number of values for a given number of digits actually increases)

wispy panther
vapid fractal
wispy panther
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@vapid fractal it's an example of a value of n (n = 756) such that 2n choose n is coprime to 105. The document uploaded contains many more.

vapid fractal
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oh dang

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so how would it help me

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could you elaborate further

wispy panther
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It doesn't help with the proof. I was just uploading it because @bleak dew seemed skeptical

vapid fractal
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oh

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so how can i prove this

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can you pls give me tips

wispy panther
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What is your current level of mathematical attainment?

vapid fractal
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undergrad

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maths

wispy panther
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The answer will probably require machinery from algebraic number theory, so you'll probably want to start heading towards that direction

vapid fractal
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hm i see

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what else

wispy panther
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Maybe?

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Anyway, it's much more difficult than my own level of math. This would be something where you might be eligible for a field's medal if solved. But you would not be solving specifically this problem, this would likely just be a corollary.

vapid fractal
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hm ok

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can we work together by any chance

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im js confused like what makes THIS PROBLEM so difficult??

wispy panther
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It is feasible that for any finite set of odd primes { a_i } there are infinitely many n such that 2n choose n is coprime to prod_i a_i

wispy panther
bleak dew
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same im first year undergrad

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this seems like magic honestly lol

vapid fractal
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im 14

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🥀

bleak dew
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💀

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im 9

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🌹

vapid fractal
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BUT I WILL BE THE ONE TO SOLVE THIS PROBLEM

wispy panther
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Don't joke about that

vapid fractal
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and change mathematics for once

vapid fractal
wispy panther
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@bleak dew gonna need you to tell me that's a joke

bleak dew
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yeah its a joke

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im just 13.5

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ask me any undergrad easy question

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first year

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i prove

vapid fractal
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guys trust me

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i will be the one to solve this, im not gonna give up !!!!

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even though people who are phd level are fumbling

wispy panther
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Best of luck.

bleak dew
vapid fractal
bleak dew
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BTW you'd be smarter than terence

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if that were true

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so maybe you have a chance

vapid fractal
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also, its not like terence has dedicated a lot of time to these particular unsolved problems

bleak dew
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yeah cos he's not stupid

vapid fractal
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if he did, then theres a possibility he could crack it

vapid fractal
bleak dew
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he's not gonna try and solve a problem that lacks the required groundwork and machinery

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It's like trying to build a GPU when all you have is sand

vapid fractal
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its not impossible though

bleak dew
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no

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its impossible

vapid fractal
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solving these maths problems isnt impossible

bleak dew
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no one knows if the theorems are even correct

vapid fractal
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just because mathematicians cant crack it, doesnt mean noone can

bleak dew
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you'd just be wasting your time, terence knows that, and knows that he only has very little time on earth, that is why he is working on so many problems that are hard but can be solvable

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there is just not the required machinery for some of these questions

pastel blade
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damn

vapid fractal
vapid fractal
bleak dew
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which isn't proven

vapid fractal
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but some problems could be revolutionary

bleak dew
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right but some problems are revolutionary and can be solved

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and we are in no shortage of those problems

vapid fractal
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k

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@wispy panther

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help a bro out

bleak dew
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i think he would agree with me

vapid fractal
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k

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shush

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now

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its his turn to speak

wispy panther
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@vapid fractal the required machinery to solve a problem can come decades or centuries after the problem is proposed, such as FLT. If Andrew Wiles lived in the 1600s no shot he would have proven it

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Doesn't matter how smart you are

vapid fractal
bleak dew
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bro you clearly aren't final year then

vapid fractal
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a genius could crack it, intelligence is profound

bleak dew
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how smart do you have to be to be this sanguine

vapid fractal
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better on being enthusiastic and driven with dedication rather than being dismissive

bleak dew
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right OmnipotentEntity can you remove this guys undergraduate role, because this guy has never done a single proof in his life

vapid fractal
bleak dew
vapid fractal
bleak dew
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yes but you have to be realistic as well,

vapid fractal
vapid fractal
wispy panther
# vapid fractal a genius could crack it, intelligence is profound

I mean, you are simply wrong. Math is developed piece by piece, and to figure out FLT we needed to develop the modularity theorem. However, these two pieces of math are only related at a deep level, and if you're looking at all of math in the 1600s you're not going to start exploring elliptic curves (which had not been seriously explored up to this point) for the answer to your problem in number theory.

bleak dew
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says a lot...

wispy panther
vapid fractal
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fairs

wispy panther
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We do review the postgrad role though.

vapid fractal
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why

vapid fractal
bleak dew
# vapid fractal why

because next year when you want the post grad and you say nonsense like this...

vapid fractal
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nonsense?

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youre being ridiculous...

bleak dew
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idek what to say to you lol. Its literally like trying to build a cpu from scratch by yourself, not gonna happen mate

wispy panther
vapid fractal
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simply underestimating intelligence, ofc there are limits and boundaries to these certain aspects in maths however, intelligence from knowledge is boundless. this could have a significant impact in the working of these unsolved maths problems. who knows, i don't think it's accurate to be making definitive statements against this belief. but ofc, i do understand what @wispy panther has been discussing about...

wispy panther
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We have processes and schematics and the CPU doesn't need to be good.

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Making pure silicon is achievable in a small lab

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Etc

bleak dew
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like ever

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because this has gotta be ragebait

wispy panther
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I think it is honestly

vapid fractal
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you are ragebait

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what does doing a proof got anything to do with this topic of conversation

wispy panther
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So I'll just stop feeding the troll, and if the troll continues saying he's hungry I'll need to take other actions

vapid fractal
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unreasonable...

vapid fractal
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hop on vc bro, takes you a month to respond and im not understanding

bleak dew
# vapid fractal unreasonable...

I should probably stop engaging as well, but literally the whole conversation was about proving questions that have stumped every mathematician, like ever

vapid fractal
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not every

wispy panther
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@vapid fractal given that a) you knowingly posted an open problem, b) you first attempted to draw people into DMs to hide this, c) gave up after you realized that I knew it was open, d) started up again when you realized I did a little work on this, it's pretty obvious that you're just trying to be disruptive.

bleak dew
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yes because it hasn't been proven lol?

vapid fractal
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honestly i dont think these erdos problems have received much attention, and bare in mind that only 41% have been solved and this is only one

wispy panther
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I've been tolerant up to this point because you were being a little bit circumspect about it.

vapid fractal
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also btw when i received this problem from someone, i didnt even know it was an open problem until ages after so i started to want other people's perspective on it

vapid fractal
wispy panther
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Anyway, I will be closing this help thread. It's an open problem. You need postgrad math to effectively attack it, and as you don't have postgrad math you'll need to work up to it.

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.close

torn parrotBOT
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Solved

Post marked as solved by @wispy panther.

Use .unsolved if this was a mistake.

vapid fractal
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.unsolved

torn parrotBOT
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Unsolved

Post marked as unsolved by @vapid fractal.

Use .solved to mark as solved.

vapid fractal
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bro

wispy panther
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Yes?

pastel blade
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what do u cover in university math

vapid fractal
# wispy panther Yes?

ill leave it as unsolved and ill come back to this channel stronger...even if i won't crack this, i'd at least have some useful research upon it...

wispy panther
vapid fractal
wispy panther
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.close

pastel blade
torn parrotBOT
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Solved

Post marked as solved by @wispy panther.

Use .unsolved if this was a mistake.

wispy panther