So you're sitting there all by yourself
and at the next table you notice there's
Freeman Dyson,
there's Armand Borel;
and they're having their own lunch very
nicely and quietly but then they start
arguing, and Borel says look give me a
prime. Dyson says 2 to the 31 minus 1.
Armand says no, I want the actual digits,
don't don't give me any exponents, I want
to actually write down the prime. So
Freeman Dyson says, all right, here's a
prime 111 111 111 111 111 1,
19 times, nineteen 1s. There's a prime for you.
And Borel says, it's
getting there but I need at least 20
digits, and they can't be all 1s. And
they think for a minute and there's some
a bit of silence
and you
cough quietly and you say, gentlemen I
think I can help you,
here's a prime for you. Twenty digits:
1 2 3 4 5 6 7 8 9
10 9 8 7 6 5 4 3 2 1.
That decimal number is a prime gentlemen.
And Borel
closes his eyes and thinks for a minute,
says
could be? Looks like you're right.
And Freeman Dyson says, well
congratulations young man,
do you know any other primes like that?
And you say well
you can look general- more generally
at what you get if you go
out up to n. So let's write down the
decimal number 1 2 3 4 5 6 7 8 9 10
10 11 12,
go all the way up to n, and we're going
to string all these digits together and
then we go down n minus 1,
n minus 2, down to 4 3 2 1. So that's a
number, rather large number if n is large,
and it depends on n. - (Brady: Can I ask a quick question?)
(Obviously)
(11 counts as 1, 1 and 12 counts as)
(1,2 and when we go back down the) - Exactly
(slide will 12 become 2, 1?)
No. 12 becomes- it stays at 1, 2. So so we
read the numbers just as they are on the
blackboard and
they go up and then they go down.
So so it goes- the the last would be
11 10
9 8 7 6 5 4 3 2 1
stop. And so
you can say when is that a prime?
(For what n?) - For what n is that a prime? Well
10 works, n equals 10 gives a prime. And
Borel says, and what's the next prime?
You say you have to go out to n equals
2446 to get another prime. And
beyond that we don't know.
So this is a pretty large prime and we
don't know anymore.
So Brady, it's a story you can tell at
parties.
And of course we would like another few
terms so we could- the sequence is not
yet- the sequence 10, 2446 is not yet
in the OEIS because we only have two
terms, it's too short. - (How many terms does)
(it have to have before you put it in the)
(Online Encyclopedia?) - There are some
sequences where one term is enough, that
are really really important. But but for
something like this
more than six digits if you can. I mean a
couple more terms would be perfect.
Officially five terms but we'll
certainly make exceptions if the terms
are big. - (Neil that prime that has n)
(equals 2446 must be a huge)
(number) - It's a huge number.
(Should we believe there are an infinite number of)
(ns this will work for?) - Yes, yes absolutely.
You can do the- do the math, you can look
at the probability that a number
like this is a prime and it's
the probability is is one over whatever
it is, and it dive- that summation
diverges. So we expect with confidence
that there'll be infinitely many. It gets
harder as the numbers get bigger and
and presumably the man who found 2446
took it some distance further but
I don't think anyone has seriously
looked into this. I call this one a
memorable prime, the one with 10 in the
middle. I'm not finished, I have another
segment. This is the most wanted prime in
the world. So you saw
the primes we just did, that you start
at 1 and you go 1 2 up to n and
then you go down n minus 1 down to 1.
But
surely the more logical question is, what
if you start at 1 and go up to n and
stop?
So let's do it: 1, that's not a prime.
12, that's not certainly not a prime;
123, well it's not a prime it's
it's 3 times 41. 1234,
obviously if it end- we're
gonna go up to n and stop, and if n is
even there's not a chance. If n is 5
there's no chance that it's a prime and
if the sum of the digits is 3
there's no chance it's a prime.
6, that's most definitely not a prime.
What about 1234567?
That looks like it might be a
prime. Well it's not, it's divisible by
127,
sorry. So we keep going, we want a prime,
we definitely want a prime. This really
is- you know, it's sort of the most
natural prime for anyone with 10 fingers.
10 11;
11 doesn't work either. 12 13;
so
let me talk about 13 for a minute, so go
12345678
1-
10 11 12 13, these are
just the digits of this number okay. And
we say, all right now that really looks
like it's a prime - it's not, it's
divisible by
113, sorry. Well all right, so
you keep going
up to n and every time you get to n you
stop and think,
and when n gets large you have to think
quite a while before you can tell
whether it's a prime or not. So
various people help me with this and
then I asked the people who run the GIMP,
the Great Internet Mersenne Prime Search
who've been looking for
Mersenne primes,
because this is exactly their kind of
problem. And they took it up, a
group of them worked on it for quite a
while. And they got up to n equal to 340,000
and they still hadn't found a prime.
And I'm afraid they've given up; because
do the math, look at the chance that a
number like this up to n is a prime,
it- roughly it looks like there's a
50/50 chance. By the time you get up to a
million you should have hit a prime.
So I would like to encourage the world
to keep going and find find this missing
prime.
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