Erdos Conjecture

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Generic Goon
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Erdos Conjecture

Postby Generic Goon » Fri Jun 19, 2009 4:03 pm UTC

So, today's comic led me to the wikipedia page on Paul Erdos, and from there to the page on "Erdős conjecture on arithmetic progressions". My question is what restrictions are there on the set in the problem? I am not too familiar with sets outside of the problem, so I am probably missing something not unique to this question, but something about sets in general.

I ask because the set: 1, 2, 1, 2, 1, 2, ... Is only positive integers, the sum of the reciprocals clearly diverges, and I don't think it has an arithmetic progression for anything greater than 2. It shouldn't be too hard to find a function to return those values, if that's one of the requirements for a set.

Anyways, thanks in advance.

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jestingrabbit
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Re: Erdos Conjecture

Postby jestingrabbit » Fri Jun 19, 2009 4:17 pm UTC

A set has no repetitions. Your sequence has lots.

So, for instance, {1, 4, 9, 25, 36,..., n2, ...} is a set, and so is {1, 2}, but not {1, 2, 1, 2, 1, 2, 1, ...}.
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Re: Erdos Conjecture

Postby t0rajir0u » Fri Jun 19, 2009 7:12 pm UTC

Generic Goon wrote:It shouldn't be too hard to find a function to return those values, if that's one of the requirements for a set.

Your question's already been answered, but what do you think a "function" is? A function doesn't have to be defined by a formula. What you've written down is a sequence, which already defines a function.

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Re: Erdos Conjecture

Postby Lul Thyme » Sat Jun 20, 2009 12:16 pm UTC

jestingrabbit wrote:A set has no repetitions. Your sequence has lots.

So, for instance, {1, 4, 9, 25, 36,..., n2, ...} is a set, and so is {1, 2}, but not {1, 2, 1, 2, 1, 2, 1, ...}.


Just a nitpick:
The more common definition is that {1, 2, 1, 2, 1, 2, 1, ...} IS a set but it is equal to the set {1,2}.

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Re: Erdos Conjecture

Postby leachboy » Mon Jun 22, 2009 3:59 am UTC

Erdos's conjecture, copied from wikipedia, is "If the sum of the reciprocals of a sequence of integers diverges, then the sequence contains arithmetic progressions of arbitrary length."

The sequence 1,2,1,2,1,2,1,2, ... does satisfy the condition that the sum of the reciprocals diverges. You can get an arithmetic subsequence of arbitrary lenth just by letting the common difference be 0 and picking as many 1's or 2's as you like.

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Re: Erdos Conjecture

Postby jestingrabbit » Mon Jun 22, 2009 7:16 am UTC

leachboy wrote:Erdos's conjecture, copied from wikipedia, is "If the sum of the reciprocals of a sequence of integers diverges, then the sequence contains arithmetic progressions of arbitrary length."


Huh?

http://en.wikipedia.org/wiki/Erd%C5%91s_conjecture_on_arithmetic_progressions wrote:Erdős' conjecture on arithmetic progressions, often incorrectly referred to as the Erdős–Turán conjecture, is a conjecture in additive combinatorics due to Paul Erdős. It states that if the sum of the reciprocals of the members of a set A of positive integers diverges, then A contains arbitrarily long arithmetic progressions.


The only link searching for your sentence turns up is an invalid subject on a radiohead forum...
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Re: Erdos Conjecture

Postby sje46 » Mon Jun 22, 2009 7:42 am UTC

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Re: Erdos Conjecture

Postby skeptical scientist » Mon Jun 22, 2009 8:35 am UTC

jestingrabbit wrote:
leachboy wrote:Erdos's conjecture, copied from wikipedia, is "If the sum of the reciprocals of a sequence of integers diverges, then the sequence contains arithmetic progressions of arbitrary length."


Huh?

http://en.wikipedia.org/wiki/Erd%C5%91s_conjecture_on_arithmetic_progressions wrote:Erdős' conjecture on arithmetic progressions, often incorrectly referred to as the Erdős–Turán conjecture, is a conjecture in additive combinatorics due to Paul Erdős. It states that if the sum of the reciprocals of the members of a set A of positive integers diverges, then A contains arbitrarily long arithmetic progressions.


The only link searching for your sentence turns up is an invalid subject on a radiohead forum...

Does it matter whether we work with sets or sequences? The two statements are obviously equivalent. If your sequence contains terms which are repeated arbitrarily often, then it contains arithmetic progressions of arbitrary length. Otherwise, the sum of the reciprocals of the sequence diverges iff the sum of the reciprocals of the set of numbers appearing in the sequence diverges, and one contains arithmetic progressions of arbitrary length iff the other does.

It is quite possible that Wikipedia changed from one version of the statement to the other, equivalent version.
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Re: Erdos Conjecture

Postby NathanielJ » Mon Jun 22, 2009 10:58 am UTC

skeptical scientist wrote:It is quite possible that Wikipedia changed from one version of the statement to the other, equivalent version.


Nah, the article has only been edited three times this year, and none of them changed "set" to "sequence" or vice-versa; they were all pretty unrelated to the wording of the opening paragraph.
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Re: Erdos Conjecture

Postby Lul Thyme » Mon Jun 22, 2009 12:45 pm UTC

Nvm

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jestingrabbit
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Re: Erdos Conjecture

Postby jestingrabbit » Mon Jun 22, 2009 5:06 pm UTC

skeptical scientist wrote:Does it matter whether we work with sets or sequences?


If you allow degenerate arithmetic progressions you might believe that the result is trivially true for all non empty sets. Obviously that's not the intention so it makes more sense to me to talk about arithmetic progressions in sets and expect the reader to understand that all the numbers in the progression must be different.
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Re: Erdos Conjecture

Postby leachboy » Tue Jun 23, 2009 12:30 am UTC

jestingrabbit wrote:
leachboy wrote:Erdos's conjecture, copied from wikipedia, is "If the sum of the reciprocals of a sequence of integers diverges, then the sequence contains arithmetic progressions of arbitrary length."


Huh?


I was quoting the wikipedia page on Paul Erdos, where it erroneously uses the word "sequence" in place of "set".

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Re: Erdos Conjecture

Postby silvermace » Tue Jun 23, 2009 1:01 am UTC



I can try to explain the new xkcd...

well Paul Erdos came up with a system of, titles for mathematicians and what not. Paul Erdos himself had the title of Erdos 0. From there, anyone who worked directly with him, would have the title Erdos 1 (very respected in the math world). Then people who worked with people who worked with Erdos would have the title Erdos 2, ect.

Now when Paul Erdos died, all hopes of obtaining Erdos 1 was lost...since he was dead. But now that the Dead are walking again, the comic shows the guy running into the math department having everyone sign some work and then getting Paul Erdos to sign it so they would all have the title of Erdos 1, a great success!

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Re: Erdos Conjecture

Postby auteur52 » Tue Jun 23, 2009 1:13 am UTC

silvermace wrote:well Paul Erdos came up with a system of, titles for mathematicians and what not. Paul Erdos himself had the title of Erdos 0. From there, anyone who worked directly with him, would have the title Erdos 1 (very respected in the math world). Then people who worked with people who worked with Erdos would have the title Erdos 2, ect.



No, Erdos did not come up with it. His friends came up with it as a joking reference to his immense amount of collaborators and publications.

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Re: Erdos Conjecture

Postby Yakk » Tue Jun 23, 2009 6:32 pm UTC

And it is typically called your Erdos number.

The alt-text is about Erdos-Bacon numbers, which are harder to get.
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Re: Erdos Conjecture

Postby skepileptic » Fri Jul 24, 2009 7:10 pm UTC

I just read that Erdos has cash rewards for certain unsolved problems of varying difficulty. Now that he's dead, you can still solve one of the unsolved problems for the prize which will be paid by someone administering the contests, but you can instead receive the original check signed by Erdos himself which cannot be cashed. So if Erdos were to come to life, you could have one of the cashed checks that had lacked the signature of Erdos signed by Erdos despite the fact that you previously opted for the cash instead of his original signed check. Who writes this stuff?

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Re: Erdos Conjecture

Postby quintopia » Sat Jul 25, 2009 3:49 am UTC

If Erdos came back to life, you could just cash one of the checks he did write. The only thing stopping you from cashing it is the fact that he's dead.

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Re: Erdos Conjecture

Postby u38cg » Tue Jul 28, 2009 2:58 pm UTC

And that the account the cheque is drawn on is long closed, and that cheques are typically only valid for six months.
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