Ruth-Aaron Pairs

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davefoc
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Ruth-Aaron Pairs

Post by davefoc » Fri Jun 18, 2004 7:30 pm

I’ve been reading a book about the life of Paul Erdos by Paul Hoffman. The book is a nice light read that is filled with interesting little snippets of math mixed in with the description of the life of a fascinating person.

One of those little snippets of math was a description of Ruth-Aaron pairs (for non-US readers see below for significance of Ruth and Aaron). These are pairs of consecutive integers for which the sums of their prime factors are equal, for instance 714 and 715.

The prime factors of 714 are 2, 3, 7 and 17. The prime factors of 715 are 5, 11 and 13. Note that the total of each is 29.

Carl Pomerance and a student discovered Ruth-Aaron pairs. The tie in to Paul Erdos is that within a week of the publication of a paper about Ruth-Aaron pairs he had called the paper’s author, Pomerance with a proof of author’s conjecture that there were an infinite number of Ruth-Aaron pairs. This resulted in the first of numerous math papers authored by Erdos and Pomerance.

This paper gave Pomerance an Erdos number of one. That means that Pomerance directly authored a paper with Erdos. An Erdos number of two means that one has authored a paper with someone who had directly authored a paper with Erdos. Erdos himself had an Erdos number of 0. There are now 485 people that have an Erdos number of one.

I did a little research on the web about Ruth-Aaron pairs. One interesting thing was that Ruth-Aaron triplets have been discovered.

See this site for a description:
http://www.primepuzzles.net/puzzles/puzz_173.htm

Question: Do you suppose that there are non-consecutive pairs of numbers for which the sum of their factors might be equal?


For non-US readers:
In America, Babe Ruth is easily the most famous athlete out of the twenties and thirties. Baseball was hugely popular at the time. And Ruth was a dominant player and character. He changed the nature of the game by his amazing ability to hit home runs. One year he hit almost as many home runs as the rest of the American league put together. He set records that lasted for years. His most famous record was for his lifetime total of 714 home runs.

Hank Aaron was the first and only man to break Babe Ruth’s record and there was considerable publicity concerning the feat when he accomplished it on April 8, 1974.

One interesting sidelight here is that in 1995 Erdos and Aaron were awarded honorary degrees at Emory University. Pomerance talked them both into signing a baseball for him.

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Brown
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Re: Ruth-Aaron Pairs

Post by Brown » Fri Jun 18, 2004 11:34 pm

davefoc wrote:Hank Aaron was the first and only man to break Babe Ruth’s record and there was considerable publicity concerning the feat when he accomplished it on April 8, 1974.
Hank Aaron was the first and only American to break the Babe's record. Sadaharu Oh hit 868 career homers in Japan.

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shemp
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Re: Ruth-Aaron Pairs

Post by shemp » Sat Jun 19, 2004 1:09 am

Brown wrote:
davefoc wrote:Hank Aaron was the first and only man to break Babe Ruth?s record and there was considerable publicity concerning the feat when he accomplished it on April 8, 1974.
Hank Aaron was the first and only American to break the Babe's record. Sadaharu Oh hit 868 career homers in Japan.
Gimme a break. That's like passing for 1,000 touchdowns in Canada. Little dinky pitchers in little dinky parks.
"It is not I who is mad! It is I who is crazy!" -- Ren Hoek

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Harry
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Re: Ruth-Aaron Pairs

Post by Harry » Mon Jun 21, 2004 8:42 am

Just wanted to gloat a little. If I every get round to finishing off writing up the damn paper I will have an Erdos number of 3... :)
Pacifism is easy when one is masochistic

davefoc
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Post by davefoc » Sat Jun 26, 2004 5:18 am

Harry wrote:Just wanted to gloat a little. If I every get round to finishing off writing up the damn paper I will have an Erdos number of 3...
Wow,
That sounds interesting. Is there any chance that you could describe the paper in words that a non-mathematician could understand? Could you tell us what the chain would be that would get you to Erdos?

Do you have some thoughts about Erdos? How do mathemeticians judge him and his contributions to the field of math?

Thanks,
Dave