- I have no doubt that others are familiar with this, but I can't recall encountering it before. It's a very simple (I like simple!) observation:

Given two or more consecutive numbers greater than 1, at least one of them will be relatively prime to the rest.

It's very easy to prove for particular cases, like two numbers (heh!), three numbers, four numbers, etc. But egad, its the proof of the general case that is elusive. (Of course!)

(If a proof were achieved, it would have prime repercussions.)

Mark - Google tells you that

Pillai conjectured ([P2]) that the following theorem holds:

Proposition 1.2. For every integer k > 16 there exists a sequence of k

consecutive integers

without a number relatively prime to all others.

The conjecture was later proven by both Alfred Brauer in [AB] and

Pillai in [P3].

You can see AB's proof here :

http://www.ams.org/journals/bull/1941-47-04/S0002-9904-1941-07455-0/S0002-9904-1941-07455-0.pdf

Maximilian

On Thu, Oct 13, 2011 at 8:43 PM, Mark <mark.underwood@...> wrote:

>

> I have no doubt that others are familiar with this, but I can't recall encountering it before. It's a very simple (I like simple!) observation:

>

> Given two or more consecutive numbers greater than 1, at least one of them will be relatively prime to the rest.

>

> It's very easy to prove for particular cases, like two numbers (heh!), three numbers, four numbers, etc. But egad, its the proof of the general case that is elusive. (Of course!)

>

> (If a proof were achieved, it would have prime repercussions.)

>

> Mark

>

>

>

> ------------------------------------

>

> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com

> The Prime Pages : http://www.primepages.org/

>

> Yahoo! Groups Links

>

>

>

> - Mark wrote:
> Given two or more consecutive numbers greater than 1, at

It's only true for up to 16 numbers and no larger amount.

> least one of them will be relatively prime to the rest.

See for example http://www.combinatorics.org/Volume_3/PDF/v3i1r33.pdf

This shows a counter example for the 17 numbers from 2184 to 2000:

for(a=2184,2200,for(b=2184,2200,if(a!=b&\

gcd(a,b)>1, print("gcd("a", "b") = "gcd(a,b));break())))

gcd(2184, 2186) = 2

gcd(2185, 2190) = 5

gcd(2186, 2184) = 2

gcd(2187, 2184) = 3

gcd(2188, 2184) = 4

gcd(2189, 2200) = 11

gcd(2190, 2184) = 6

gcd(2191, 2184) = 7

gcd(2192, 2184) = 8

gcd(2193, 2184) = 3

gcd(2194, 2184) = 2

gcd(2195, 2185) = 5

gcd(2196, 2184) = 12

gcd(2197, 2184) = 13

gcd(2198, 2184) = 14

gcd(2199, 2184) = 3

gcd(2200, 2184) = 8

--

Jens Kruse Andersen - Thank you Maximilian and Jens. I never would have guessed that very interesting result. I should have gone higher in the length of consecutive numbers I examined. But no, I had to rush to a conclusion, even neglecting The Google!

So, I just Googled it, and see a Prime Curio for the number 10:

"The largest number N such that any set of N consecutive integers contains at least one integer relatively prime to all other integers in the set. [Rupinski]"

This looks like it was misplaced, and should be a Curio for 16 instead.

Thanks again,

Mark - Mark wrote:
> So, I just Googled it, and see a Prime Curio for the number 10:

Yes, the quoted curio for 10 is wrong. It is at

>

> "The largest number N such that any set of N consecutive integers

> contains at least one integer relatively prime to all other integers in the

> set. [Rupinski]"

>

> This looks like it was misplaced, and should be a Curio for 16 instead.

http://primes.utm.edu/curios/page.php?number_id=126&submitter=Rupinski

There is already a right curio for 17 here:

http://primes.utm.edu/curios/page.php?number_id=15&submitter=Pillai

"In any set of k < 17 consecutive integers there is always one which is

relatively prime to all the others. [Pillai]"

--

Jens Kruse Andersen