Erdős–Woods Numbers - Numberphile

Featuring Dr James Grime on Erdős–Woods Numbers. More links & stuff in full description below ↓↓↓

NUMBERPHILE

Video by Brady Haran and Pete McPartlan

I find it deeply satisfying to know that Erdő means woods or forest in Hungarian.

[illogicmath]

The prof's shocked face at 9:52 when Brady asked about overlapping sequences was priceless!

[buzzzysin]

Around 10:00 "Great question! I don't know!" This is the essence of discovery and why I love this channel

[mmorizes]

9:44 I am always in awe of Brady's ability to ask such beautiful questions that either I was thinking of, or after hearing, can't believe I hadn't thought of first.

[A-V]

@09:09 I think the most surprising thing was seeing that the two mathematicians, Erdös & Woods, have the Hungarian version and the English version of the same basic name :)

[stanleydodds]

To answer one of the questions, yes, sequences do overlap (and sufficiently long sequences will always overlap with some other sequence(s)). Note that for any start point and end point, we can add any multiple of the product of all required prime factors to both ends, and it will still be a valid sequence of the same length - call this product of primes a period of this length (there may be other sequences within each period, but it is sufficient to know some finite period exists). Since there are infinitely many Erdos-Woods numbers, they are unbounded, and are eventually larger than this period and larger than the minimum start point of the sequences of that period. That means any sequence of sufficiently large length must overlap with at least one of these sequences which occur with sufficiently small period.

[TigburtJones]

Maths and science are the lifeblood of imagination. Imagination is the soul of creative human endeavor.
Never stop creating.

[xdkristof]

make the sequence 2 numbers long, you won't have to cross anything off because there will be nothing to cross off, ez win

[spenjaminn]

5:01 even funnier how there has already been a numberphile video about the number 2184

[builder]

“And now I’m gonna make a sequence until I get bored”
me as a math kid be like

[matthewkendrick]

All of them are divisible by one, hope this helps!

[AexisRai]

4:30 Thoughts:
Some notation:
Seq(a, b) = [a]{a+1, ..., b-1}[b]
a and b in N, a < b
S = {a+1, ..., b-1}.
For what nontrivial Seq(a, b) does every element of S share a factor with a or b?
(If b-a = 1, S = {}, which trivially passes.)



Yes, the first intuition is that primes in S are bad because they won't share factors with a or b.[1] But there's also the idea that a and b shouldn't be prime either, because then they won't share factors with elements of S.

[1] (Technically if a < p < b with p prime and let b = pk, then you could cancel p - but then because "there is always a prime between n and 2n", there would be a different prime q between b and p, which b would not be a multiple of.)

Really it seems like what you want is for a and b to have _a lot_ of prime factors, with the intuition that the more primes you have, the more of S you can "cancel out".

But then there's the problem that a and a+1 won't share any common prime factors, and b and b-1 won't share any common prime factors. (You can verify this.)
So for a+1 to get canceled, it must be by a factor that b contributes, and for b-1 to get canceled, it must be by a factor that a contributes.

I'll stop there and see what they say next.
(edit: a word)

[jeremybuchanan]

"... starting at b, which is a completely different number ..." plausible t-shirt worthy quote imo

[vegalyra]

"I've thought of something funnier than 24... 35!"

[mattc]

Really happy I tried this one before watching the answer, it was fun!

[orterves]

It's always fascinating to me that integers, and primes specifically, are so fundamental and there are so many unanswered questions about them. Questions that can be described to children yet unanswered by even the greatest mathematical minds

[SovernGaming]

This number sequence comes up in an episode of Mr. Robot when solving a puzzle. Fun lil cameo

[IrishEye]

Love these videos. Someone with the brain the size of a planet looking into Primes got bored and found this. Marvellous.

[Matthew-bufg]

Side Note: the only reason I knew that the numbers 2, 184 and 2, 197 had 13 as factors is because of the Numberphile Video titled: "Why 1980 was a great year to be born... but 2184 will be better"

Extra side note: that video will boom in popularity next year

[maze]

Minor correction: In Hungarian, the letters ö (short) and ő (long) are different letters. In the video Paul Erdős is shown with the wrong (short) letter, since he is written with the long one (as in the video title)