Well, I'd like to talk a little bit about baseball, and in particular, I have two numbers.
714
and 715.
Now, one reason I think many mathematicians
really love baseball is because it's,
at the base of it, when you watch it on television,
it's a fairly boring game.
It's slow-moving and it's filled with numbers!
The sports casters always talk about batting averages.
They talk about on-base percentages.
They talk about records, like the most home runs someone has ever hit.
In their career or in a particular year or whatever.
Babe Ruth, who is one of the greatest baseball players of all time,
he played for the Boston Red Sox,
and then very famously, or infamously, was traded to the New York Yankees.
Over his career, he hit 714 home runs.
This was a gigantic record who towered over everybody else,
and, it was thought that this record would never be beat.
Fast-forward to, you know, later in the 20th century
1960s and 70s
and we have Hank Aaron, who played for the Atlanta Braves,
who was closing in on Babe Ruth's record.
It really captured the nation's attention and, in fact, there were lots of overtones involved with this.
Babe Ruth played for baseball at a time when baseball was segregated.
All of the players were white.
After World War II, baseball became integrated.
Hank Aaron was African-American.
He was met with many racist attacks for trying to beat the record of the white baseball player,
Babe Ruth.
And through all of these really ridiculous and stupid and vicious attacks, he kept his dignity and kept his cool.
And he just, you know, kept on doing what he does best, which is play baseball.
And so, it transcended just the
the-the sports...moment.
And-and in fact,
the numbers 714 and 715
carried a weight with them, that anybody
in the US at that time would know what those numbers were referring to
without even mentioning, Ruth, Aaron or baseball.
So I'm a mathematician, I'm watching the game,
715 is hit, there's a ceremony on the field, the game is stopped for the ceremony,
and it's a wonderful moment, and then I'm thinking,
"Okay, these are numbers. I'm a number theorist. What can I tell you about the numbers 714 and 715?"
So the first thing I try, is I try to factor them.
They're not hard to factor.
714, its prime factorization is 2 times 3 times 7 times 17.
715, that's also easy to factor.
715 is 5 times 11 times 13.
And what do we notice but, for these primes that are involved?
They are all of the primes up to some point.
So in other words, if I multiply 714 times 715
and factor that into primes, I get every single prime appearing up to 17.
Isn't that interesting?
Two consecutive numbers. Their product is the product of all of the primes up to some point.
Now, you can ask
"How special is this property?"
"Does it ever happen again, or does it happen for smaller numbers?"
Well, you can see that
it does happen for smaller numbers. For example, 5 and 6 work.
14 and 15 work. You start searching for larger examples than 714 and 715, good luck!
You are likely not to find it and
it's not known if there are any larger examples, but we can't, we can't prove it,
but we strongly think there are no larger examples.
Brady: "How? Has that been like a search of note? Has someone put these into a computer and let it run for a few weeks or...?"
Well we tried this at, I was a professor then at the University of Georgia
and we tried this. I and a few colleagues
and I don't remember how far we got, but,
it, the num- when you take a product of consecutive primes,
up to some level and multiply them together, these numb- these products grow exponentially.
In fact, I believe it's a consequence of the ABC conjecture,
that, there should only be finitely many examples.
Anyway, the next day, when I went into work,
I spoke to a colleague, and I said
"Can you find an interesting property of 714 and 715?"
And, he,
after a few minutes, found the same property that I did, but then he went into his class and asked his class
"Can anybody in this class find an interesting property of 714 and 715?
and one of the students found another property that's also amazing.
What if I change these times signs into plus signs?
For- you know, that's perverse, but let's do it!
So we add 2 plus 3 plus 7 plus 17. That adds up to
29.
And, if I do it for 715 and add them together,
that also adds up to 29.
And here, we were able to find more examples,
probably infinitely many examples of this, although, although rare.
We wrote an article for a non-serious
mathematical journal. To give you an idea of how non-serious it is, the name of the journal is,
'The Journal of Recreational Mathematics'.
In this article, we had lots of jokes,
and we may, you know, had fun with all of these interesting properties of 714 and 715.
We also had a conjecture, and the conjecture was that, these pairs of two consecutive numbers,
whose primes add up to the same thing,
this is a 'Ruth-Aaron' pair,
and we conjectured, they thin out. The technical name is 'have density zero'.
Anyway, it turns out that this whole thing, which started as a joke,
this article was read by Paul Erdős
who is one of the most famous mathematicians from the 20th century ..
He wrote a letter to me saying that he can prove
that the 'Ruth-Aaron pairs' thin out.
And he wanted to come to Georgia to discuss it with me.
This is incredible!
Here I am, a young beginning professor, writing a joke article for the 'Journal of Recreational Math',
and then one of the greatest mathematicians of the century should write me a letter saying he wants to come and discuss it.
So he did! He came. He discussed it with me. We wrote a joint paper.
I started learning the tricks of his trade,
and he mentored me, and we ended up writing over 20 joint papers before he passed away in the 90's.
It completely changed my career. And the funny finale for this story is that
in 1995 I believe,
Emory University in Atlanta gave Paul Erdős an honorary degree.
Erdős invited my wife and I for the party before, the evening before,
to celebrate the recipients of the Honorary degree,
and it turns out that there was another person there also getting an honorary degree, at the same time.
This was Hank Aaron.
When I got there, I just couldn't believe that Hank Aaron,
who is the reason, for some sense, that I met Paul Erdős in the first place, was, was there in the same room with Paul Erdős.
So, I went up to Aaron and introduced myself and I said that his baseball feat
completely changed my career, as a mathematician.
Well, he looked at me in a very puzzled way,
like maybe he had just met a crazy person.
As usual with him, he was very magnanimous and calm and pleasant, and I introduced him to
Erdős and they had a nice conversation,
and there's a photograph of them sitting at a table together.
One of the Emory professors who was there, knew that Aaron was coming and
had brought some baseballs for Aaron
to sign, which is sort of a customary thing to do, to have a great star athlete sign a baseball as a memento.
So he gave me one of these baseballs
and I had Aaron sign it, and then I had Erdős sign it.
And, you know, people have Erdős #1 if they have a joint paper with Erdős.
Well, I argue that, Aaron should have Erdős #1. He doesn't have a joint paper, but he does have a joint baseball.
Brady: "What happened to the baseball? Where is it?"
Well, one of my students gave me a little plastic box to put it in, so preserve it,
however, not ever having had people sign baseballs before, I had- they signed it with a felt-tip pen,
and within about a year or two the signatures had faded,
and now, maybe if you were a great forensic expert, you could maybe find
those signatures on the baseball, but, I -they're essentially invisible.
It just looks like a new baseball.
publish in this area, would you like to review for us?
And Peck said, sure.
So Peck started reviewing papers, and he reviewed one of Kleinman's papers,
which was a little incestuous.
P was again, George Purdy, of Cincinatti, I'm the G, C was Chung, Fan Chung, K was Kleiman, E was Erdős. It was good to have a vowel.