A Satisfying Number Theory Proof

[@JohannNeumann-ic3gy]

2 months ago I didn't know you. Now I'm a full time viewer. Keep up the great content!

[@DrBarker]

Thank you!

[@drooga81]

love your videos - amazing how clearly you speak.

[@sr6424]

Another interesting video. I did my my puzzle evening last night. I create a puzzle based on last week’s video (Median of Medians) - it wend down really well!

[@DrBarker]

That's great to hear!

[@AntoshaPushkin]

Does it work for numbers other than 10? It feels like it could work for numbers up to some limit, and then stop working

[@NoNameAtAll2]

if my calculations with python code are correct, then you can't make it with less than 14 numbers (aka only with 2,3,5,7,11 primes) first consequtive non-coprimes are 2184...2200

[@NoNameAtAll2]

you just have to check for - gaps between k-smooth numbers (aka those not divisible by first k primes) - check gaps up to k-th primordial (multiplying first k primes together) - gap must be bigger than k-th prime and - gap must have 2 numbers divisible by k-th prime 11 is first to have large enough gap, but only one divisible by 11 13 gives the 2183+1...2183+17

[@AntoshaPushkin]

@@NoNameAtAll2 awesome, thanks for checking!

[@NoNameAtAll2]

​​@@AntoshaPushkin okay, after a bit more of analysis I confirmed that there's no earlier solution with bigger primes that there's no solution with 16 or less in a row and that 2186..2200 is essentially the only solution for 17 in a row (up to addition of a multiple of 2*3*5*7*11*13 and a minus sign)

[@NoNameAtAll2]

It is called "stapled interval". OEIS sequence: A090318

[@BedrockBlocker]

You're amazing.

[@rsivaraman1729]

Indian mathematician S.S. Pillai proved that among any set of k consecutive integers for k less than 17, one among them is coprime to every other.

[@Tehom1]

It needs to be "positive integers" because zero isn't coprime to anything.

[@Dshado]

I have a technicality I want to address. This is only true for N>1.

[@Umbra451]

Technically I suppose you could go all the way down to starting with -1 before you hit a snag. Past that you get additional common factors of -1 that complicate things, and as far as a cursory stackexchange-search goes, prime-ness for negative numbers does not seem to be well-defined.

[@armanavagyan1876]

Thank U UR VIDEOS are pretty interesting)

[@robertveith6383]

Write in English. This is not partial texting.

[@AzharLatif-d4z]

Where is the Proof ? You have only demonstrated the properties of composite numbers with common factors. What is the point which leads to a conclusion.

[@pietergeerkens6324]

It's in there - just obfuscated. By Distributive Law (or, alternatively, Extended Euclidean Algorithm) any common factor of two numbers is also a common factor of the smaller and the difference. Thus any common factor must be a factor of a Natural Number from 1 to 9 (the only possible values for the difference between two of our list values). That, I believe, sets up the rest of the video more cleanly.