What's a quasiperfect number? Well, a
perfect number is a number that's equal
to the sum of its proper divisors, like
six. Its divisors are 1, 2, 3
and 6, but that's not a proper divisor.
1 plus 2 plus 3
equals 6, so 6 is a perfect number.
A quasiperfect number? Well that's a
number where the sum of the proper divisors
is one more than the number itself,
that's a quasiperfect number! Only thing
is nobody's ever found any.
But it's still an open question as to
whether or not there are
any; and so as a high school student I
actually worked on that problem and
proved that
if there were any quasiperfect numbers
lurking out there they had to at least
satisfy certain properties.
(Brady: Of all the things, like)
(there's all these glamorous things like)
(prime numbers and proper perfect numbers,)
(why did you pick)
(this obscure little one that has this- ) - Ha!
Well I was a student at a
National Science Foundation summer
program in mathematics that-
that was held at Hampshire
college in Massachusetts.
And it was a wonderful six week program
and learned lots of math;
and amongst other things, one afternoon, I
went to the Hampshire college library,
which didn't have that all many books on
mathematics but did have
some book on number theory. And I pulled
out this book
and I found that - I even remember - there
was there was a description of
quasiperfect numbers and
some result that had been proved about
them by someone called Sierpiński.
And we'd been talking about perfect
numbers and other things and I was
struck by that result and I wondered,
could I do any better? And it turned out
I could do somewhat better!
And I ended up writing a Westinghouse
prize contest paper about that; the
Westinghouse is now the-
the Intel prize and it ended up winning
the thing.
But mostly what was cool to me was I
could see something in a textbook
and find that I could extend, even by a
little bit, about
what's kind of a relatively unimportant
subject
all of human knowledge by some tiny
little bit. And once you learn that
that's kind of addictive.
Oh I confess I don't even remember all
the details about what it was, it's now been
uh- let's see, you know, more than 35 years
since I proved that; but it was a set of
constraints on
what the structure of the number was, uh
I think it had to be
one more than a square and some other
things and how many prime factors -
and the truth is I haven't followed
quasiperfect numbers since then.
It's probably not a really really
important question per se,
although the question about the
distribution of divisors and their sums
that's a good and important question in
in number theory. But it was more that it
was a totally understandable, tractable
question
that a high school student could
understand, that wasn't hard to pose
the definition of it; and to realize that
you could pose a definition in a
sentence
and that all the mathematicians in the
world still didn't know if there were any,
meant that there were just an infinite
number of really cool mysteries out
there;
they were accessible in some way. So,
you know, I loved mathematics for that. As
a high school student I loved
mathematics, I did my
undergraduate degree in pure mathematics
at Princeton and my PhD
in pure mathematics at Oxford; and
you know just loved it, I think so many
people love mathematics because
within your mind there is a world that
that you can create and experiment in
and find the regularities in and the
laws of - without needing a laboratory,
it's all accessible. - (Brady: It doesn't seem the)
(default position for a high school)
(student to read about that)
(and then think: let's nut out a proof.)
(They're more likely to read that and)
(think, oh I wonder if I can find one? To)
(start taking stabs in the dark. But you)
(went and made like a)
(a proper structured mathematical proof?)
Well, you know, I think
I think the cool thing- this particular
National Science Foundation program
that that I went to as high school
student had as its motto
'Find Patterns Make Conjectures Prove
Things'. I think that sums up what
mathematics is about.
And we gave emphasis to all of those; so
I went there as a student then I went
back and taught for three summers as a
faculty member at it while I was in college,
teaching high school students about it.
And I think all three have an equally
important role in mathematics; finding
pattern,
people- non-mathematicians get the sense
that mathematics is this
really dry and boring thing, somebody
gives you a problem and asks you to
prove it or calculate a number or
something. Of course what mathematicians
really do
is they look for the regularities in the
world and the discoveries that get made
are the most interesting part of
mathematics. Then when you see a pattern
you don't quite know how to describe it
and phrase it and the act of phrasing a
conjecture:
I believe that this pattern I- I'm seeing means
that there's an infinite number of twin
primes, for example.
Or that there are no solutions- that,
that all the solutions to a certain type
of equation; which primes can be
written as the sum of two squares, have a
certain form.
Learning how to take a pattern and turn
it into a rigorous conjecture,
that's also an important work of
mathematicians. And then finally when you
have a rigorous conjecture
figuring out how you could possibly
prove it. And so
I think we learned most of all that that
was what mathematics was about and
therefore
when you saw something, and there was a
conjecture about it, you wanted to think
about how you might prove that.
So yeah it was actually perfectly
natural for, you know, math kids. I was a
math kid in New York City,
I was a lower middle class kid from
Brooklyn. I had the opportunity to go to
Stuyvesant high school in Manhattan, one
of the- one of New York City's absolutely free
public but exam schools
for math and science. And
well I- I loved it. I got to Stuyvesant
high school in ninth grade, discovered
there was this math team, people meeting
at eight o'clock in the morning
at Stuyvesant high school, coming in an
hour before classes to get together to
do problems. And so the math team quickly
became a home for me, I came in an hour
early, went to the math team every day
for the course of four years, eventually
became the captain of that math team, and
in those those ancient
days before computers and the internet
or everything there was a special
trove of problems that
we had to work on and they were all
stored in 'The Shopping Bag'.
The Stuyvesant high school math team had
'The Shopping Bag'
filled with problems and it was in the
care of the captain of the math team
that the shopping bag was kept.
(Brady: Because you won this really prestigious prize)
(because of these numbers, I'm imagining)
(you feel some kind of-)
(do you still find any kind of attachment)
(to it? And like would you love to find)
(out if there is one or have you just)
(moved on now and it's like- ?)
Well, I mean it would be cool to find out
but I've- I've moved on since. There are
other things I'm fascinated by.
[Music]
I think what was- what was wonderful
about the
summer high school program, was wonderful
about the Westinghouse contest,
what was wonderful about the Stuyvesant
high school math team
was much more the community of people
who cared.
What it really meant to me was there was
a life in this, you could actually make a living,
you could actually have a community, you
could actually have friends and colleagues
who cared about deeply intellectual
problems. And you know growing up where I
did in the outskirts of Brooklyn, not
everybody
was like that. And so for a student to
discover that there's a world of people
who care about such things
and live for such things; whether it's
mathematics or biology
or mathematics and biology - things I do
today - that was the real eye-opening thing; and
if quasiperfect numbers was a way to
get there that was great
but I don't necessarily have an
attachment to them but they were a
wonderful guide along the
along the trail. - (You don't feel like you)
(owe them one?)
No, no I think, I think we're square.
(Do you think there's one out there?)
(What's your- what's your gut tell you?)
Oh, goodness
um; well-
It's really tough, I don't have a good
gut feeling on it. I mean,
you know, I'd like to say there aren't
any, because none have been found,
but I don't think there's a tremendously
good reason to believe there aren't any.
[Preview] The iPhone - which is gorgeous, I love my
iPhone - seems so important and directly valuable;
but all the fundamental physics and the
electronics, the computer science that
makes an iPhone
possible - they're crucial to the
existence of an iphone.
And Steve Jobs would have said that, and
Tim Cook at Apple will say that;
that the beautiful things you encounter
in your daily life,
whether it's an iPhone or a drug that'll
cure a disease...