I watched a youtube video not long ago about some set of pairs of numbers named, of course, after a mathematician. The interesting bit was that it was conjectured that the three pairs mentioned in the video, that satisfied some rules, were the only three possible pairs that satisfied those rules.
I thought it was rather interesting, but now I can’t remember the name of the conjecture, or find the video again, so I’m hoping this rings a bell with some of the teeming millions.
Numberphile took up Brown numbers (n,m) where n! + 1 = m[sup]2[/sup]. Only three such pairs are known, e.g. (7,71): 7! is one less than 71[sup]2[/sup]
I thought there was a chance I could get an answer here even with my vague recollection. I’m shocked it was that quick. Thanks a bunch! 
I knew it would be Numberphile just reading the title.
While many of their videos are pretty good, some are horrid and get basic facts wrong, munge the explantion, etc. Take the “expert” in this video. Gets the name and location of Muḥammad ibn Mūsā al-Khwārizmī wrong, etc. And 3 is divisible by 2. Just not evenly divisible (or not within the Integers, or with no remainder, etc).
So take what they say sometimes with a grain of salt.
You’re arguing against the standard mathematical definition of divisible. Example here.
It’s not really meaningful to talk about divisibility in the context of fields (like the rationals or reals) since everything is divisible by everything (except zero). i.e. If a and b are reals the statement a divides b doesn’t tell us anything about the relationship between a and b since the statement is logically equivalent to a is not zero.
Yeah. I’d like to see an example of divisible used the way you seek to define it here ftg. If all that exists is examples of people contesting the useful standard meaning, I’m afraid I’ll have to take your post with a grain of salt. 
But this is NOT a video addressed to Mathematicians. It’s addressed to the general public. The general public is not nearly as clear about such matters as Mathematicians!
These sort of videos should be overly careful on the use of terminology.
So giving cites, etc., doesn’t matter due to the intended audience. And referring to “fields” and such really gets into terminology not remotely suitable.
Think about the purpose of the video, please.
Many of the Numberphile videos are interesting and informative, but the quality is very uneven. Many of them waste much time on tedious and irrelevant detail.
I just watched one called “The Prime Problem with a One Sentence Proof.” The prime problem is Fermat’s difficult “Christmas Theorem” and the “one-sentence proof” is this one by Zagier. The lecturer said he’d spend ten sentences instead of one sentence but all ten sentences were devoted to the most trivial part of Zagier’s single sentence, clarifying nothing. I hope one of the Board’s mathematicians appears and fully explains Zagier’s sentence for us!
The existence of numbers like Mill’s constant is very cute but the Numberphile narrator gushes about how special this magic number is. In fact, once you understand why it works, you realize such numbers are “a dime a dozen.” Mill’s constant is simply defined to be the smallest of the infinitely many such numbers.
(BTW, Brady Haran contacted and cited me for a Numberphile — my 15 seconds of fame — but despite several e-mails he obstinately focused on a misunderstood detail.)
Agree with ftg. At a bare minimum they ought to use terms like “3 is not evenly divisible by 2”.
When teaching simplified stuff for simplified uses to simplified people you need to simplify very carefully. Fighting ignorance can easily turn into fomenting ignorance.
I can’t find it just now, but in the last 6-ish months we had a thread (or digression in thread) on the lasting harm done to people’s understanding of, IIRC, physics by the simplifications taught in the early elementary years.
Better if teachers, including video teachers, include more explicit caveats about the limitations of their generalizations. e.g. “This isn’t always true everywhere, but while at least now while we’re talking about X you’ll see that Y is true.”
So teach people the wrong definitions of words? If a is divisible by b, then a/b is an integer. That’s just what the word means. Otherwise, the word is pointless, since any number is divisible by any other number, save 0.
I’m also not too impressed by claims they got the history wrong, when they are a maths channel. The people involved are working mathematicians (most with doctorates), not historians. And, if you pointed it out in the comments, they will correct mistakes anyways.
Now, if you have a problem with them not explaining things in a way that you get it, that’s fair. I admit, I wasn’t satisfied with that one sentence proof video, either. But that’s one of thousands of videos.
But Dr. Brady Haran does great work.
These sort of videos can be overly careful on the use of terminology,
because the audience is the general public, they should stick to common words like
“divisible”. It was clear the entire video was about integer concepts such as
-1 and "Integer ^ 2 " and x!.
I never said I was against using the right terminology.
If fact, I am 100% in favor of using the right terminology.
If …
you explain the terminology!
“Divisible” is not an intuitive concept to most people. Remember, the audience might be people who grew up on calculators so integers are nothing special or 3rd graders who haven’t been exposed to much Number Theory.
I was a college professor. You’d be amazed what even upper division Computer Science majors don’t know about basic Math principles.