Kind: captions
Language: en
this is a video about the oldest
unsolved problem in math it dates back
2,000 years some of the brightest
mathematicians of all time have tried to
crack it but all of them
failed in the year 2000 the Italian
mathematician Pier Georgio Oda Freddy
listed it among four of the most
pressing open problems at the time
solving this problem could be as simple
as finding a single number so
mathematicians have used computers and
checked numbers up 210 to the power of
2,200 but so far they've come up
empty-handed why do you think this
problem has captured the imaginations of
so many mathematicians it's
old it's
simple it's
beautiful what what else could you want
so the problem is this do any odd
perfect numbers
exist so what is a perfect
number well take the number six for
example you can divide it by 1 2 3 and
six but let's ignore six because that's
the number itself and now we're left
with just the proper divisors if you add
them all up you find that they add to
six which is the number itself so
numbers like this are called perfect you
can also try this with other numbers
like 10 10 has the proper divisors 1 2
and 5 if you add those up you only get
eight so 10 is not a perfect number now
you can repeat this for all other
numbers and what you find is that most
numbers either overshoot or undershoot
between one and 100 only 6 and 28 are
perfect numbers go up to 10,000 and you
find the next two perfect numbers 496
and
8,128 these were the only perfect
numbers known by the ancient Greeks and
they would be the only known ones for
over a thousand years if only we could
find a pattern that makes these numbers
then we could use that to predict more
of them so what do these numbers have in
common well one thing to notice is that
each next perfect number is one digit
longer than the number that came before
it another thing they share is that the
ending digit alternates between six and
8 which also means they are all
even but here's where things get really
weird you can write six as the sum of 1
+ 2 + 3 and 28 as the sum of 1 + 2 + 3 +
4 + 5 + 6 + 7 and so on for the others
as well they are all just the sum of
consecutive numbers and you can think of
each additional number as adding a new
layer and so these create a triangle
which is why these numbers are called
triangular numbers also every number
except for six is the sum of consecutive
odd cubes so 28 is 1 cubed + 3 cubed
496 is equal to 1 cubed + 3 cubed + 5
cubed + 7 cubed and
8,128 is equal to 1 Cub + 3 Cub + 5 Cub
+ 7 Cub + 9 Cub all the way up to 15
cubed but here's the one that really
blows my mind if you write these numbers
in binary 6 becomes 11 1 0
and 28 becomes 111 0 0
496 becomes 1 one11 0 0 and
8,128 you guessed it it is also a string
of ones followed by a series of zeros so
if you write them out they are all just
consecutive powers of two
what now around 300 BC uclid was
actually thinking along similar lines
when he discovered the pattern that
makes these perfect numbers take the
number one and double it you get two now
keep doubling it you get 4 8 16 32 64
and so on now starting from 1 add the
next number to it so 1 + 2 = 3 if that
adds up to a prime then you multiply it
by the last number in the sequence to
get a perfect number so 2 * 3 = 6 the
first perfect number now let's keep
doing this add 1 + 2 + 4 and you get
7even which is again prime so multiply
it by the last number four and you get
28 the next perfect number next add 1 +
2 + 4 + 8 = 15 but 15 isn't Prime so we
continue add 16 to get 31 this is prime
so you multiply it by 16 and you get 4
196 the third perfect
number now you can keep doing this to
find bigger and bigger perfect numbers
and using this we can rewrite the first
three so 6 = 1 + 2 * 2 ^ 1 and 28 = 1 +
2 + 4 * 2^ 2 and 496 = 1 + 2 + 4 + 8 +
16 * 2 ^ 4 where the first term is prime
but there's a more convenient way to
write this still take any sum of
consecutive powers of two so 2^ of 0
which is 1 plus 2 1 + 2 2 all the way up
to 2 the n minus1 and now because you
don't know n you don't know what that is
equal to but it will be equal to
something so let's call that t now
multiply this whole equation by two so
you get 2 1 + 2 2 all the way up to 2
the n and this is equal to 2T if you now
subtract the first equation from the
second
almost all the terms will cancel out and
you're left with t = 2 N
-1 so you can replace this whole series
with one less than the next power of two
so 6 becomes 2^ 2 - 1 * 2 1 28 becomes 2
Cub - 1 * 2^ 2 and 496 becomes 2 5 - 1 *
24 do you see the
pattern this number is always one more
than this so if we call this P then
ucl's formula that gives a perfect
number is 2 p - 1 * 2 pus1 whenever this
is
[Music]
prime now because you're multiplying it
by 2 the P minus1 which is even this
will always give an even
number uid had found a way to generate
even perfect numbers but he didn't prove
that this was the only way so there
could be other ways to get perfect
numbers including potentially ones that
are
odd 400 years later the Greek
philosopher nikus published introductio
arithmetica the standard arithmetic text
for the next thousand years in it he
stated five conjectures statements he
believed to be true but did not bother
actually trying to
prove his conjectures were one the nth
perfect number has n digits two all
perfect numbers are even three all
perfect numbers end in six and8
alternately four ucd's algorithm
produces every even perfect number and
five there are infinitely many perfect
numbers for the next thousand years no
one could prove or disprove any of these
conjectures and they were considered
facts but in the 13th century Egyptian
mathematician ibben phis published a
list with 10 perfect numbers and their
values of P three of these perfect
numbers turned out not to be perfect at
all but the remaining ones are the fifth
perfect number is 8 digits long which
disproves ncis's first conjecture and
the next thing to notice is that both
the fifth and sixth perfect number end
in a six so that disproves nicolas'
third conjecture that all perfect
numbers end in a six or eight
alternately two conjectures were proven
false but what about the other
three two centuries later the problem
reached Renaissance Europe where they
rediscovered the fifth sixth and seventh
perfect
numbers so far every perfect number had
ukids form and the best way to find new
ones was by finding the values of P that
make 2 to the pus One Prime so French
polymath Marin meren extensively studied
numbers of this form in 1644 he
published his results in a book
including a list of 11 values of P for
which he claimed they corresponded to
primes numbers for which this is true
are now called meren primes of his list
the first seven exponents of P do result
in primes and they correspond to the
first seven perfect numbers but for some
of the larger numbers like 2 to the 67
minus one meren admitted to not even
checking whether they were Prime to tell
if a given number of 15 to 20 digits is
prime or not all time would not suffice
for the test
meren discussed the problem of perfect
numbers with other luminaries of the
time including Pierre de FMA and Renee
de cart in 1638 dayart wrote to meren I
think I can show that there are no even
perfect numbers except those of uid he
also believed that if an odd perfect
number does exist it must have a special
form it must be the product of a prime
and the square of a different number if
he was right these would easily have
been the biggest breakthroughs on the
problem since you 2,000 years earlier
but deart couldn't prove either of those
statements instead he wrote as for me I
judge that one can find real odd perfect
numbers but whatever method you use it
takes a long time to look for
these around a 100 years later at the St
Petersburg Academy the Prussian
mathematician chrisan goldbach met a
20-year-old Math Prodigy the two stayed
in touch corresponding by mail and in
1729 goldbach introduced this young man
to the work of FMA at first he seemed
indifferent but after a little more
prodding by goldbach he became
passionate about number Theory and he
spent the next 40 years working on
different problems in the field among
them was the problem of perfect numbers
this Prodigy's name was Leonard
Oiler oer picked up where dekart had
left off but with more success in doing
so he made three breakthroughs on this
problem first in 17732 he discovered the
eighth perfect number which he had done
by verifying that 2
31-1 is prime just as meren had
predicted for his other two
breakthroughs he invented a new weapon
the sigma function all this function
does is it takes all the divisors of a
number including the number itself and
adds them up so take any number say six
sum up all its devisers and you get 12
which is twice the number we started
with and this will be true for for all
perfect numbers the sigma function of a
perfect number will always give twice
the number itself because the sigma
function includes the number as one of
its divisors now this may seem like a
small change but it ends up being
extremely powerful so let's look at a
few examples take a prime number like
seven now because it's Prime you can't
rearrange it into a rectangle therefore
the only divisors are one and the prime
itself so Sigma 7 is 1 + 7 which is is
equal to 8 now to keep things easier to
follow we'll just stick to the numbers
but what if instead of 7 you had 7 cubed
well again the sum of the divisors is
really simple it's just 1 + 7 + 7^ 2 + 7
cubed now let's use it on a different
number say 20 the sum of its divisors is
1 + 2 + 4 + 5 + 10 + 20 which equals 42
but you can also write this as 1 + 2 + 4
* 1 + 5
and this is what really makes the sigma
function so powerful if you have a
number that is made up of other numbers
that don't share factors with each other
then you can split up the sigma function
into the sigma functions of the Prime
powers that make it up so Sigma of 2^2 *
Sigma 5 is equal to Sigma 20 and since
any number can be written as the product
of prime Powers you can split up the
sigma function of any composite number
into the sigma functions of its prime
powers
with his new function in hand Oiler
achieved his second breakthrough and did
what dayart couldn't he proved that
every even perfect number has ukids form
this uclid Oiler theorem solved a 1600
yearold problem and proved nikolas's
fourth conjecture math historian William
Dunham called it the greatest
mathematical collaboration in
history but Oiler wasn't finished yet he
also wanted to solve the problem of odd
perfect numbers so so for his third
breakthrough he set out to prove
decart's other statement that every odd
perfect number must have a specific
form because if an odd perfect number
does exist you know two things first n
is odd and second Sigma of n equals 2 N
now any number n you can write as a
product of different prime numbers and
each prime can be to some
power so let's take that and put it into
oil's Sigma function so you get Sigma of
n n equals Sigma of all of those primes
to their powers which equals 2 N but
since all of these factors are primes
you can actually split up the sigma
function into the sigmas of the
individual Prime Powers now one thing to
notice is that if you have a prime
number raised to an odd power for
example 7 ^ of 1 then the sigma function
will be even because 1 + 7 = 8 you'll
always get an even number because odd
plus odd is even if the prime number is
instead raised to an even power like 7^
squar then the sigma function returns an
odd number Sigma of 7 squar = 1 + 7 +
7^2 which equal 57 because odd plus odd
+ odd equals odd so if you have the
sigma function of an odd prime raised to
an odd power it will give an even number
if instead it's raised to an even power
you get an odd
number and this is where oil's genius in
sight comes in because here on the right
side You've Got 2 * n where n is an odd
perfect number and two is even well what
that means is that on the left side
there must only be one even number
because if there were two even numbers
you could factor out four but that means
you should also be able to factor out
four on the right side which you can't
because n is odd and there's only a
single two here so only one of these
sigmas here can give an even number
which means that there is exactly one
prime that is to an odd power and all
the others must be to an even power just
as decart had
predicted now Oiler refined the form a
bit more and showed that an odd perfect
number must satisfy this
condition but even Oiler couldn't prove
whether they existed or not he wrote
whether there are any odd perfect
numbers is a most difficult question for
the next 150 years very little progress
was made and no new perfect numbers were
discovered English mathematician Peter
Barlo wrote that Oilers 8 perfect number
is the greatest that ever will be
discovered for as they are merely
curious without being useful it is not
likely that any person will ever attempt
to find one Beyond
it but Barlo was wrong mathematicians
kept pursuing these elusive perfect
numbers and most started with Mer's list
of proposed primes the next on his list
was 2 the 67 minus 1 so far meren had
done an excellent job he had included
Oilers eighth perfect number while
avoiding others like 29 that turned out
not to lead to a perfect number but 230
years after meren published his list
Edward Luca proved that 2 67- 1 was not
prime although he was unable to find its
factors 27 years later Frank Nelson Cole
gave a talk to the American mathematical
Society without saying a word he walked
to one side of the Blackboard and wrote
down 2 67 - 1 equal 147 quintillion 573
quadrillion
952 tril 589
b676
m412
n27 he then walked to the other side of
the Blackboard and multiplied
193,000 7,72
1 times 761 b838
m257
287 giving the same answer he sat down
without saying a word and the audience
erupted in
Applause he later admitted it took him 3
years working on Sundays to solve this a
modern computer could solve this in less
than a
second from 500 BC until 1952 people had
discovered just 12 meren Primes and
therefore only 12 perfect numbers the
main difficulty was checking whether
large meren numbers were actually Prime
but in 1952 American mathematician
Raphael Robinson wrote a computer
program to perform this task and he ran
it on the fastest computer at the time
the swac within 10 months he found the
next five meren primes and so
corresponding perfect numbers and over
the next 50 years new meren primes were
discovered in Rapid succession all using
computers the largest mercen prime at
the end of 1952 was 2 to the^ of
2,281 minus1 which is 687 digits long by
the end of 1994 the largest mercen prime
was 2 to the^ of
859 433 minus 1 which is
28,700 long since these numbers were
getting so astronomically large the task
of finding num meren primes became more
and more difficult even for
supercomputers so in 1996 computer
scientist George Waltman launched the
great internet meren Prime search or
gimps gimps distributes the work over
many computers allowing anyone to
volunteer their computer power to help
search for meren primes the project has
been highly successful so far having
discovered 17 new meren primes 15 of
which were the largest known primes at
that time and the best part if your
computer discovers a new meren Prime
you'll be listed as its Discoverer
adding yourself to a list that includes
some of the best mathematicians of all
time there's even a
$250,000 prize for the first billion
digigit
Prime in 2017 Church Deacon John Pace
discovered the 50th meren prime by using
gimps the number 2 to the 77
m232 N17 minus 1 is more than 20 3
million digits long and it was also the
largest known prime at the time to
celebrate this achievement the Japanese
Publishing House Nan rosha published
this book the largest prime number of
2017 and all it is is that number spread
over 719 glorious
Pages it's wild the size of this font is
so tiny the book quickly Rose to the
number one spot on Amazon and sold out
in 4
days a year later the 51st meren Prime
was discovered it's two to the 82 M
5899 33 minus 1 and this number has 24,
862,5kg
a book but in some way it's nice that
there's this physical artifact that like
has the number if ever we lost all the
prime numbers you know someone could
find this book be like here's a big one
as of today this is still the largest
known prime and since numbers of this
form grow so rapidly the largest meren
prime is almost always the largest known
[Music]
prime computers have been incredibly
successful at finding new meren Primes
and their corresponding perfect numbers
but we've still only found 51 so far so
you might suspect that there are only a
finite number of them which would mean
that nicolas' fifth conjecture would be
false that there aren't infinitely many
perfect
numbers but that might not be the case
the lenstra pomerance Wagstaff
conjecture predicts how many meren
primes should appear based on how large
p is now this is the actual data the
conjecture performs remarkably well but
more importantly it predicts that there
are infinitely many meren primes and so
infinitely many even perfect numbers the
mercen primes are just so large and rare
that they take a lot of time and
Computer Resources to
find but a conjecture is not a proof and
up until this day this problem shares
the title of oldest unsolved problem in
math with the other open problem do any
odd perfect numbers exist
the easiest way to solve this problem is
by finding an example so maybe we could
just check different odd numbers and see
if one of them is perfect that's exactly
what researchers tried in 1991 by using
a smart algorithm called a factor chain
they were able to show that if an odd
perfect number does exist it must be
larger than 10 to the^ of 300 21 years
later Pascal Oak and Michael Ral raised
that lower bound to 10 to the 1
,500 with recent progress pushing that
number up to 10 to the
2,200 with numbers that large it's
unlikely that a computer will find one
anytime soon so we'll need to get smart
what would a proof look like like how
could we actually prove this I I think
the main idea that people have been
trying to approach this problem with is
coming up with more and more conditions
odd perfect numbers have to satisfy it's
called this web of conditions where it
has to have
10 prime factors now that we know and
maybe thousands of non-distinct prime
factors and has to be bigger than 10 to
the 3,000 and it has to do all these
different things and we hope that
eventually there's just so many
conditions that can strain the numbers
so much that they can't exist since
Oiler mathematicians have kept adding
new conditions to this web but so far it
hasn't
worked but there might be another path
when decart was looking for for odd
perfect numbers he came across 198 B 585
m576 189 which you can Factor as 3^ 2 *
7 2 * 11 2 * 13 2 *
22,021 put this into Oilers Sigma
function and you find it is equal to 2 *
the original number in other words it is
perfect that is if 20221 were Prime but
it's not because it is equal to 19 S *
61 and filling that in shows that it is
not perfect numbers like this that are
very close to being odd perfect numbers
are called spoofs spoofs are a larger
group of numbers so odd perfect numbers
share all properties of spoofs and then
a few extra ones and the goal is to find
properties of spoofs that ultimately
prevent them from being odd perfect
numbers for example one condition of odd
perfect numbers is that they can't be
divided by
105 so if you find that spoofs must be
divisible by 105 then this would prove
that odd perfect numbers can't exist in
2022 pace neelen and a team at BYU found
21 spoof numbers including decart's
number and while they discovered some
new properties of spoofs they didn't
find any that rule out odd perfect
numbers so how large would an odd
perfect number have to be they don't
exist
you don't think odd perfect numbers
exist no they don't
exist I wish they did that'd be really
cool if if there was this this one
gigantic odd perfect number out in the
universe they they don't exist no how
are you convinced that they don't
exist there is uh there's something
called a heuristic argument where it's
not a proof so if we had a proof we'd be
done it's just an argument from okay we
think primes occur this often of this
type and you put that those pieces of
information together and you think okay
on average how many numbers should be
perfect this argument which was made by
Carl pomerance predicts that between 10
to the 2200 and infinity there are no
more than 10 to the- 540 perfect numbers
of the form Nal pm^
s with odd perfect numbers theistic says
we should expect any uh We've searched
high enough now that we think we have
enough evidence they shouldn't exist
anymore my understanding is this
heuristic argument it also predicts that
there are no large perfect numbers even
or
odd so it that's
true so there's a downside to yeah
there's a downside because it says there
shouldn't be large even perfect number
numbers and we actually expect there to
be infinitely many and so okay so why do
I believe theistic in this case and not
this case you're right uh am I being
hypocritical about that there are other
aspects you can add on to the heuristic
and make it stronger let me put it that
way but you're right it's not a proof
for now this is still the oldest
unsolved problem in math Oiler was right
when he said whether there are any odd
perfect numbers is a most difficult
question
so are there any applications of this
problem I I can say
no now many people may think that if
there are no applications to the real
world then there's no point studying it
why should anyone care about some old
unsolved problem but I think that's the
wrong approach for more than 2,000 years
number Theory had no real world
applications it was just mathematicians
following their curiosity and solving
problems they found interesting proving
one result after another and building a
foundation of useless mathematics but
then in the 20th century we realized
that we could take this foundation and
base our cryptography on it this is what
protects everything from text messages
to government secrets whenever you have
a group of people put their minds
towards a problem something good is
going to come out of it if it's only if
it's only at the beginning this doesn't
work okay well as Edis said I I learned
9999 ways of not making a light bulb
eventually I got a good way to do it
it's the same with math you you you have
a problem and you throw your mind at it
and others do too and you come up with
new ideas and eventually something good
comes from that process Einstein's
general relativity was built on non-
ukian geometries geometries that were
developed as intellectual Curiosities
without foresight of how they would one
day change the way we understand the
universe how many people do think are
working on the problem of perfect
numbers right now I'd guess around 10
people uh currently have papers in the
area 10 to 15 if if you're a high
schooler and you just love mathematics
and you think I want to problem to think
about this one's a great problem to
think about and you can make progress
you can figure out new things yeah don't
be scared hundreds of people have
thought about this problem for thousands
of years what can I do you can do
something why should you do math if you
don't know that it will lead anywhere
well because doing the math is the only
way to know for sure you can't tell in
advance what the outcome will be like
this problem might turn out to be a dud
we might solve it and it might not mean
anything to anyone or it could turn out
to be remarkably helpful the only way to
know for sure is to try
in today's world it often feels like
you've got to choose between following
your curiosity and building real skills
you can apply but the truth is it's
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