There is a great quora answer for this one: https://www.quora.com/How-do-you-find-the-positive-integer-s...

>quora answer for this one

that quora article is written by Alon Amit and fwiw, TFA quotes Alon Amit, so his thoughts are ex post.

now that's peak Quora right there!

I really miss those years when Quora was a decent place. it went downhill when they decided to aggressively monetize it.

the same happened to poe.

I heard they use LLMs to generate questions and answers to those questions to help drive traffic, which is why no matter what insane bullshit you type into google, you will probably get a quora page about it.

Something something dead internet theory.

I saw a reddit thread from sub about {specific retailer} in {specific geographical region} where the OP ends with ChatGPT's classic "Would you like some more variations of this text?" response and the whole damn thread was other users happily replying on topic, totally missing that OP was clearly a bot because they were probably bots too. But hey, they all get karma, all get activity and look they're members of {geographic region} so their opinions on future political threads in {region} are totally legit right?

But yeah, something something dead internet theory indeed.

Yeah, yeah, they just mumble mumble diophantine mumble mumble elliptic curve mumble mumble Weierstrass form mumble mumble like your boring old uncle at Christmas. GET TO THE ANSWER!

   154476802108746166441951315019919837485664325669565431700026634898253202035277999
   36875131794129999827197811565225474825492979968971970996283137471637224634055579
   4373612677928697257861252602371390152816537558161613618621437993378423467772036

On a related tangent, when I was teaching my younger kids about math and helping with homework, I would often rewrite things as a formula, or rewrite the formula itself once they got to that point. However instead of things like x I would use things like fluffy cloud, star, etc. They thought it was annoying but it still kept them interested and have said they've done the same when helping their peers. It's easy to forget what it was like learning these abstractions, and it was important to show that x was nothing special, it could be a sun, it could be the phrase "total number of kittens"

> it could be the phrase "total number of kittens"

There is this minimalism present in math culture, while I sort of understand it, when tossing and mixing formulas it helps to have everything as small as possible. but later, when it is published it really sucks for readability, "Okay here is an item doing some heavy lifting in this formula what is it for? hell if I know some joker labeled it 'φ' "

I like to joke, If you think programmers are bad at naming thing you should see the mathematicians, they take a perverse pride in their inability to name things.

The worst are programs derived directly from a math paper, if your variable holds the correlation coefficient call it that. we have thousands of years of language and labels we can use to share our ideas with others. don't encrypt it and call it "rho".

One "fun" aspect to this is that there's not just 1 but 13 extra copies of the English alphabet in Unicode that are there specifically for mathematicians to name stuff in formulae and equations.

And with the exceptions of Fraktur and script, free fonts that are already sans-serif and monospace generally do not make the glyphs visually distinct from the actual English alphabet characters similarly weighted and slanted.

Which gives the sighted people a taste of what the sight-impaired people have to suffer when people abuse these code points as ways to write in italics or boldface, because the screen readers correctly obey the important note in Unicode Technical Report 25 that these letters do not combine to make words, and read them out as mathematical formulae where "d" is multiplied by "o" and then by "g".

> don't encrypt it and call it "rho".

These programmers were not hazed enough in their fraternity days. It was not drilled into them that p is rho and that rho is p. Whichever one they choose was always wrong and punished with pushups.

I tried giving this to ChatGPT. (Just by uploading the image to the base OpenAI interface.) I expected to either (a) have the model already know the question and give the right answer, (b) hallucinate an answer or (c) refuse to engage with the problem at all.

Instead, this happened:

https://chatgpt.com/share/682cce62-c53c-8003-be2c-2929395868...

Basically, the model confidently outputs a guess, then calculates it, determines it to be incorrect, and repeatedly tries again, even repeating the same guesses over and over. It does not recognize any symmetry and acts like a completely unstructured agent. In the end, the model vehemently asserts there to be no solutions to this puzzle. I really did not expect this and will update my beliefs accordingly if the models behave as badly with future puzzles.

Here's gemini https://g.co/gemini/share/ab287b25648f

I also asked Chat GPT o3 and it thought for 11.5 minutes! https://chatgpt.com/share/682d0993-db4c-8004-a66c-3908ef7203...

I'm impressed. There certainly are no "reasonable" solutions, where I am arbitrarily defining reasonable as numbers that humans can work or understand in their heads.

Isn't there a version of ChatGPT that connects to Wolfram Alpha? Did you try that one?

> A guy by the name of Sridhar Ramesh

Sridhar is a pro follow, you don't see many people with PhDs in both math and shitposting.

I love this genre! I've started calling it “Dantzig Sniping” and here’s my own one: https://x.com/TheOisinMoran/status/1298305686082744320

Some more context and related ones here: https://x.com/TheOisinMoran/status/1299124512240398336

I thought you named it for Gdańsk (Danzig) and was wondering what’s got sniped there :)

How does one discover problems with this property?

It's year 2025, why doesn't the author use actual fruit emoji for variable names?

When trying to understand complex C codebase I've often found it helpful to rename existing variable as emojis. This makes it much easier to track which variables are used where & to take in the pure structure of the code at one glance. An example I posted previously: https://imgur.com/F27ZNfk

Unfortunately most modern languages like Rust and JS follow the XID_Start/XID_Continue recommendation (not very well-motivated imo) which excludes all emoji characters from identifiers.

wouldn't writing a parser of sorts that would replace emojis with a valid alphabetical string identifier be trivial?

You're right that writing a preprocessor would be straightforward. But while you're actively editing the code, your dev experience will still be bad: the editor will flag emoji identifiers as syntax errors so mass renaming & autocompletion won't work properly. Last time I looked into this in VSCode I got TypeScript to stop complaining about syntax errors by patching the identifier validation with something like `if (code>127) return true` (if non-ascii, consider valid) in isUnicodeIdentifierStart/isUnicodeIdentifierPart [1]. But then you'd also need to patch the transpiler to JS, formatters like Prettier, and any other tool in your workflow that embeds their own version of TypeScript...

[1] https://github.com/microsoft/TypeScript/blob/81c951894e93bdc...

Here's Gemini's (brute force?) solution to the problem in C# using fruit emoji variables:

https://imgur.com/a/cC5QPH0

Nice! Not going to work, but appreciate the emoji.

UPD. This code is like that picture with Disney princesses. The closer you look, the scarier it gets

Yeah, don't deploy this to production :laughing_emoji:

I do use a lot of emojis in logs and status windows and in my comments. I'd not really thought about them in my code, but now I'm going to try to think of a way they could be used to improve code in a sensible way.

While current year is 2025, the year the language was made in probably isnt

Good or not, many languages choose not to support emoji, even newer ones.

The reasons are complicated but, many people prefer unicode normalization so that different forms of what appear to be the same word are considered the same word. People argue whether or not this is important but it can certainly be argued that it would be frustrating to get an error like

    let café = 1;
    café += 1;  // error, unknown identifier 'café'

The error happens in non-normalizing languages because those to idenifiers are not the same unicode.

But, choosing a normalization affects emoji as well. Worse, when new ones are added the normalization rules can change.

The level of blind trust that English speakers put on non-ASCII characters support always throw me off, knowing username on Windows 11 still has to be short ASCII sequences. Surely it's not 2010 anymore and you only have to recreate the user account rather than clean re-installing Windows, but still.

Non-ASCII comments in a source code can be scary enough sometimes, unless it's for an all-Unicode system like Android or something HTML based.

I had an idea the other day to use 64-bit character codes. Then each code can be interpreted as an 8x8 bitmap. Every character would be guaranteed to have a unique bitmap representation. The bitmaps wouldn't bet used for rendering, of course -- but they could be used as a fallback if your font does not define a character. Anyway this would somewhat avoid the problem you describe because two characters that look the same visually would have the same value. Nothing I'll ever implement of course, just a thought experiment.

As someone who has worked with 8×8 fonts I can report that you'd have some surprising problems with that idea. Not only would you have problems with there being two distinct forms of letters like "a" and "g", making things over-unique; and not only is it tricky to differentiate forms that actually are not the same, because they are from two different alphabets (especially the 13 extra "mathematical" alphabets); but it's actually quite difficult to make pre-composed forms in that amount of space.

8×8 is a tight squeeze, and 16×16 works a lot better. But that would make your approach vastly more space hungry than a normalization approach using the actual Unicode code points.

* https://github.com/jdebp/unscii/tree/2.1.1f

The early pages of the Swift docs show you can use emojis as variable names, and iirc that was the very first page in the Swift 1.0 handbook. I have a theory that the language design was originally motivated by emojis, because there were also interesting choices around strings, like originally having no ".length" method.

Here: https://docs.swift.org/swift-book/documentation/the-swift-pr...

Easter egg: The example is named dogcow, after a 90s Mac icon, designed by Susan Kare, which later became a small mascot: https://512pixels.net/dogcow/

Regarding .length: Effectly that is just the result of Unicode, there is no one-to-one equivalent between characters code points, the code units in an encoding and the resulting grapheme clusters. That is in effect a result of the complexity of the world's alphabets, including Emoji.

Yeah I get the justification for the lack of .length, but they eventually added it for a good reason too, which is that anyone calling that doesn't really care and those who do care can use something more specific.

The other aspects of strings are also centered around things being of uncertain length, like how it's O(n) to take the nth character of a string, and how there are rather complicated objects involved in taking substrings. There's a lot more thought and resulting complexity than other languages' default strings. And yes a few languages use extended grapheme clusters, but I feel like emojis were the real motivation.

I appreciate the deep dive into the depths of number theory and strange graphs, but what is supposed to be tricky/confusing about the original apple/banana/etc puzzle?

Is there supposed to be some easily confused bit so people argue or just be easy so everyone rushes to show off?

(I got 10, 4, 2; maybe it confused me?)

The “trick” is that there are only three bananas in the last bunch and four in the others. Likewise there is a single coconut in the last equation. It appears this could be interpreted as 1 + 10 + 3.

That's actually half a coconut. The smaller one is ambiguous. Is it supposed to be another half or a quarter?

Using other constants in place of the ‘4’ can lead to some _really_ gigantic smallest solutions: https://observablehq.com/@robinhouston/a-remarkable-diophant...

Ah, I really really love the dumb but magnificent on-demand loading of the 120-million-digit-long numbers in a nice little textarea!

I had fun with that! I'm happy someone appreciates it.

S3's support of the Range: header makes it possible to do that sort of thing remarkably cheaply and efficiently.

I remember the problem when it first appeared, we had a good laugh at the number theory seminar :)

The "srsltid" in the query parameters is unnecessary: https://artofproblemsolving.com/community/c2532359h2760821_t...

Dumb question, but how do you know it's smallest?

Not dumb at all, Alon Amit addressed it in the Quora post linked above.

https://www.quora.com/How-do-you-find-the-positive-integer-s...

The theory of elliptic curves goes amazingly deep. Scratching the surface slightly more, to fill in the article's "ignore the labels like 2P":

Intersecting the curve with lines the way the author does is, perhaps shockingly, a commutative group operation, known as point addition. You define this operation by saying that the three points A,B,C on a line sum to zero: that is A+B+C=0 or in other words, A+B = -C. Reflecting across the curve's line of symmetry is negation (there's an alternative definition that extends to curves without reflection symmetry). Combining the two defines an operation A+B which adds two points on the curve and gives a third one: draw the line from A to B, intersect it with the curve to find a point -A-B (using the same type of formula given in this article) and then reflect it to get A+B. This addition operation obviously commutes (meaning, the line between A and B is the same as the line between B and A), but surprisingly it also associates and you get a group operation.

(For math olympiad nerds out there: so the union of a conic and a line is also a bivariate cubic equation. You can carry out the same "addition" operation there. Again the operation clearly commutes. But it also associates! This is basically Pascal's theorem.)

The theory of elliptic curves is also the basis of elliptic curve cryptography. In that case, instead of the curve being over the reals, all the calculations are done mod some prime p, which destroys structure based on continuity and prevents the numbers from becoming too large. There are a bunch of subtleties here but the key is that you can still straightforwardly compute addition in this context, with basically the same formulas. Then from addition you can get n*A, both for small integers n (eg 5*A = A+A+A+A+A) but also for large n (e.g. 2*A = A+A; 66*A = 2*2*2*2*2*2*A + 2*A). This "scalar multiplication" operation is a one-way operation for appropriately chosen curves: it's easy to calculate n*A from (n,A), but as far as we know it is hard to calculate n from (A,n*A) ... at least if you can't build a large quantum computer.

This gives you mix of easy operations (eg, addition and scalar multiplication) plus problems believed to be hard, which is a great starting point to build cryptography. There are a lot of important technical details, though fewer than with the new lattice schemes.

(Further comment on the olympiad thing: so you can do this with conic+line too, extending from addition to multiplication and using the same formulas mod p. But it's not as good: it's not hard to find a projection that sends the line to infinity and the conic to the unit circle or similar, and then the group operation becomes equivalent to multiplying the points' coordinates as complex numbers or similar. If the group operation is equivalent to multiplication, then scalarmul becomes equivalent to exponentiation, which ends up being in Fp or Fp^2 depending on some Legendre symbol or other. Exponentiation is still potentially secure: it's basically classical Diffie-Hellman. However, more attacks are known on exponentiation in Fp or Fp^2: the attacks don't outright break it but you need p to be much bigger.)

Edit: unescaped stars make italics.

Once upon a time I was good at math -- like, win state-wide competitions, get a scholarship good. It's sad that I can easily follow the path laid out here, but I am nowhere close to being able to lay out this course on my own.

Nobody can do this on their own. You don't get to do this by having a great aptitude for high school maths and then being exposed to one or two more concepts that just intuitively click - which is enough to be competitive at the level you describe, though not enough to make the US Olympiad team.

You get the level of understanding necessary to create this blog post by already having worked through several textbooks and papers on number theory, elliptic curves, etc. So you don't do it on your own, you do it with help from Poincaré and Euler and Diophantes and whoever else.

Based on your description, you absolutely have the aptitude to do this if you pursued maths further academically. And also by having great teaching and explaining skills - that part might be harder to learn.

The picture says 95% are unable to solve this - I wonder what the correct percentage is. Closer to 99.999% I would guess, based on your explanation.

Originally, more like 99.999995%

When I'm teaching congruence or the Gauss method to solve equation systems, I tell my students: "You can solve the midterm alone or with the ghost of Gauss sit on the side chair."

I like that. "Standing on the shoulder of giants" is a fairly common way to phrase this, and it captures truth as well, but "on a team with the greatest mathematicians of history" is pretty fun too.

Pity I'm by far the weakest link in such team-ups. :)

> point being, mathematics is not a team game. but everything else is??

Sorry, it’s early, and I can’t tell if you’re being sarcastic. I hope you are? Mathematics is very much a team game.

i think based on the username they seem to be looking for validation through upvotes. Perhaps assuming this place is like a nerdier reddit.

actually, reddit is a mainstream hacker news

to me old.reddit.com will always be reddit.com

Ah I never changed from old.reddit, I am sure the current version is entirely unrecognisable. It throws me sometimes when a comment mentions a users pfp, because wtf? Reddit doesn't have user profiles with pictures.

So the meme was a crap-post and never "supposed" to be solved anyway (in human terms?)

At least it was a problem with a known solution obtainable by a standard procedure in not too many steps. Could've also picked a deceptively simple Diophantine equation where it is unknown whether it even has any solutions or not: https://thehighergeometer.wordpress.com/2021/07/27/diophanti...

I think the real joke is that the average Facebook user, with no higher math, might be tricked into trying to solve it, and independently rediscover methods for solving diophantine equations and elliptic curves, all so that they can prove that they're in the 3% by commenting what the strawberry equals.

Hah yeah. But probably more like 0.03% (no kidding)