Kind: captions
Language: en
this is the most dangerous problem in
mathematics one that young
mathematicians are warned not to waste
their time on it's a simple conjecture
that not even the world's best
mathematicians have been able to solve
Paul erush a famous mathematician said
mathematics is not yet ripe enough for
such
questions here's how it works pick a
number any number seven good choice okay
we're going to apply two rules if the
number is odd we multiply by three and
add 1 so 3 * 7 is 21 + 1 is 22 if the
number is even we divide by two so 22 ID
2 is 11 now we keep applying these two
rules 11 is odd so we multiply by 3 33
and add 1 34 even divide by 2 17 odd
multiply by 3 51 add 1 52 even divide by
2 26 still even divide by 2 13 odd so we
multiply by 3 39 add 1 and that's 40
which is even so we divide by 2 20 IDE
by 2 10 IDE by 2 5 odd multiply by 3 15
add 1 16 divide by 2 that's 8 and then 4
2 and 1 now one is odd so we multiply by
three and add one which equals four but
four goes to two goes to one so we're in
a loop and the lowest number is one now
the conjecture is this every positive
integer if you apply these rules will
eventually end up in the 421 Loop this
is commonly called the colets conjecture
after German mathematician lther collets
who may have come up with it in the
1930s but the problem has many origin
stories and many names it's also known
as the ulam conjecture kakutani's
problem the we's conjecture hass's
algorithm the Syracuse problem and
simply 3 n + 1 why is 3x +1 so famous
among professional mathematicians maybe
it's not famous but Infamous in the
sense that if someone actually admits in
public that they're working on it then
there's something wrong with
them the numbers you get by applying
3x+1 are called Hailstone numbers
because they go up and down like
hailstones in a thundercloud but
eventually they all fall down to one or
at least we think they do you can think
of the numbers as representing the
height above the ground in meters so a
number like 26 would start 26 M above
the ground and if you apply 3x + 1 it
rises up as high as 40 m and in total it
takes 10 steps to get to one so 10 is
called its total stopping time but take
the very next number 27 and it bounces
around all over the place in fact it
climbs all the way up to
9,232 as an altitude that is higher than
Mount Everest before it too falls back
to the ground in total it takes 111
steps for 27 to get down to one and end
up in the 421 Loop the path that
different numbers take vary so widely
even numbers right next to each other so
how do you even start to make progress
on this problem well honestly
mathematicians struggled people just
decided that this was something invented
by the Soviets to slow down us science
and it was doing a good job at it cuz
everybody's sitting there twiddling
their thumbs and making no progress on
this trivial thing that you can tell
school children Jeffrey legarias is the
world Authority on 3x+1 the first time I
met him I was a a senior in college and
he pulled me aside and he said don't do
this don't work on this problem if you
want to have a career here do not start
spending time writing about this or
publishing any papers about this do real
math for a while to establish yourself
Alex kovich didn't listen he and Yakov
Sinai looked at the paths of the hail
stone numbers were there any patterns
well obviously all of them end up at one
but what about the paths they take to
get there the pattern is
Randomness here is the sequence of a
large number chosen at random the graph
Peaks and then drops so low that you
can't really see what's happening at
this scale but if you take the logarithm
you find this Wiggly graph with a
downward Trend it looks kind of like the
stock market on a bad day and this is no
coincidence both are examples of
geometric brownie in motion that means
if you take the log and remove the
linear Trend the fluctuations are random
it's like flipping a coin each step if
the coin is heads the line goes up Tails
it goes down 3x + 1 is just like the
random Wiggles of the stock market over
long enough periods the stock market
tends to Trend upwards while 3x + 1
Trends
down another way to analyze 3x+ 1 is by
looking at the leading digit of each
number in a sequence here are the
Hailstone numbers starting with three as
the seed and we can count up how many
numbers start with a one how many start
with a two how many start with a three
and so on to make a histogram we can do
the same thing for the SE that starts
with four that's a short one and for the
sequences that start with five six and
seven again for each sequence we're just
counting up how many numbers start with
each digit 1 through n and adding that
to our
histogram if you keep doing this for
more and more numbers eventually the
histogram settles into a stable pattern
for the first billion sequences you'll
find one is by far the most common Le
leading digit 30% of all numbers start
with 1 around 17 1 12% start with two
12% start with three and the frequency
decreases for higher digits fewer than
5% of all the numbers start with nine
now this pattern is not unique to 3x + 1
it actually comes up everywhere from the
populations of countries to the value of
companies all the physical constants and
the Fibonacci numbers just to name a few
the distribution is known as benford's
law and it is even used to detect fraud
if all the numbers on your income tax
forms obey benford's law then well
you're probably being honest if not you
may be hiding something in elections
benford's law can be used to spot
irregularities though you have to apply
it correctly benford's law works best
when the numbers involved spans several
orders of magnitude as they do for 3x +
1 but benford's law can't tell tell us
whether all numbers will end up in the 4
to1 Loop or not for that we need a
different sort of analysis now at first
glance it seems strange that when you
apply 3x + 1 all numbers should end up
at one I mean consider that there are
the same number of odd and even numbers
but odd numbers are more than tripled
while even numbers are only cut in half
therefore it seems like every sequence
on average should grow not shrink but
but here's the catch every time you
multiply an odd number by three and then
add one it will always become an even
number and that means the next step is
to divide by two so odd numbers are not
actually tripled by 3x + 1 they're
increased by a factor of about 3 over2
I'm neglecting the plus one because it's
insignificant for large numbers and
three halves is actually the most an odd
number can grow in one step think of the
path from one odd number in a sequence
to the next odd number after multiplying
by three and adding one you have an even
number and 50% of the time dividing by
two brings you back to an odd number but
a quarter of the time you can divide by
four before you get to the next odd
number so for a quarter of numbers the
next one in the sequence will be 3/4s of
its initial value an eighth of the time
you can divide by eight before getting
to the next odd number and a 16th of the
time you can divide by 16 and so on if
you take the geometric mean you find on
average to get from one odd number to
the next one you multiply by 3 over4
which is less than one so statistically
speaking 3x + 1 sequences are more
likely to shrink than grow take 341 for
example multiply by 3 and add 1 you get
1,24 which you can divide by two and
then divide by two again and again and
again and again 10 times in total until
you're down to one one way to visualize
these Paths of numbers in 3x + 1 is
simply to show how each number connects
to the next one in its sequence this is
called a directed graph it looks like a
tree or a series of little streams that
flow into each other if the conjecture
is true it means that every single
number is connected to this graph every
tiny stream all the way out to Infinity
eventually flows into the massive River
of 4 2
1 some mathematicians have modified this
visualization by rotating the graph at
each number anticlockwise if it's an odd
number and clockwise if it's even and
then you end up with a structure that
looks like a coral or seaweed and by
adjusting the degree of rotation for odd
and even numbers you can create these
beautiful organic looking
shapes now there are two ways the
conjecture could be false there could be
a number somewhere some seed that starts
a sequence of numbers that grows to
Infinity for whatever reason it doesn't
obey the same numerical gravity as all
of the other numbers another possibility
is there exists a sequence of numbers
that forms a closed loop all the numbers
in this Loop would be unconnected to the
main graph but thus far no Loop or
sequence that shoots off to Infinity has
been found and not for lack of trying
mathematicians have tested by Brute
Force every single number up to 2 to the
68 that's 295 quintillion 147
quadrillion 95 trillion 179 billion 352
million
825,000 quintilian numbers and none of
disproves the conjecture in fact given
this information mathematicians
calculate that any Loop other than 421
must be at least 186 billion numbers
long so it seems pretty likely that the
conjecture is true but this doesn't
prove it one way mathematicians have
attempted to prove it is by making a
scatter plot with all the seed numbers
on the x-axis and a number from each
seed's sequence on the y- AIS now if you
can show that in every 3x +1 sequence
there is a number that is smaller than
the Original Seed well then you have
proven the cat's conjecture because
whatever number you pick you know it
will at some point get smaller and that
smaller number as a seed also gets
smaller and so on down to one meaning
the only way any sequence can end is in
the 421 Loop this has not yet been shown
but in 1976 rho teras was able to show
that almost all Callet sequences reach a
point below their initial value in 1979
this limit was reduced with almost all
numbers going to less than x to ^ 869
and then in 1994 it was further lowered
2 less than x ^
7925 in this case the term almost all
numbers has a technical mathematical
definition it means that as the numbers
you're looking at go to Infinity the fra
action that end up under the curve goes
to
one then in 2019 one of the world's
greatest living mathematicians Terry ta
was able to show 3x + 1 obeys even
stricter criteria he showed almost all
numbers will end up smaller than any
arbitrary function f ofx so long as that
function goes to Infinity as X goes to
Infinity but the function can rise as
slowly as you like so log X is an
example or log log X works too or log
log log log X what this means is for
almost all numbers you can guarantee
that there is an arbitrarily small
number somewhere in its sequence in a
public talk he gave in 2020 Terry to
said this is about as close as one can
get to the callets conjecture without
actually solving it this is an
impressive result but it's still not a
proof so why can't we prove the
conjecture true could it be because it's
not true I mean everyone is trying to
prove it true which means almost no one
is looking for counter examples it
happened to me just uh two years ago
where there was there was something I
was trying to prove that I was trying
for 3 years to
prove and I couldn't get it to work and
then I found a counter example and then
I realized what the correct statement
should have been and then a month later
I proved the correct statement maybe we
should be spending more energy looking
for counter examples than we're
currently spending
I mean remember how the number 27 grows
all the way to
9,232 here is a plot of seed numbers up
to 10,000 with the largest number
reached for each seed plotted on the y-
axis the y-axis stops at 100,000 but not
all numbers can be shown at this scale
the seed
9,663 for example climbs as high as 27
million and as yet no one has proven why
a number couldn't just shoot off to
infinity and it would take only one to
disprove the
conjecture or some set of numbers could
be part of a loop not connected to the
main graph as far as we know there is
only one Loop 4 21 but something strange
happens if you include negative numbers
applying the same 3x + 1 rules as before
there is not one Loop not two Loops but
three independent Loops of numbers and
they start at low values like -7 and
-5 why should there be disconnected
loops on the negative side of the number
line but not on the positive side now
one of the most convincing pieces of
evidence supporting the conjecture is
Terry to's proof that almost all numbers
have a number in their sequence that is
arbitrarily small but proving that
almost all numbers obey this criteria
isn't the same thing as proving that all
numbers
do how many numbers between one and 100
are perfect squares the answer is 10 so
10% of numbers up to 100 are perfect
squares how many numbers between 1 and a
th are perfect squares the answer is 31
so only 3.1% of the numbers up to a
th000 are perfect squares and the higher
you go the smaller this percentage
becomes such that in the limit you could
say almost all numbers are not perfect
squares the fraction of numbers that are
not perfect squares goes to one as X
goes to infinity and yet we know there
are an infinite number of perfect
squares and we know exactly where they
are now we've tested by brute force all
numbers up to 2 to the 68 and they all
obey the call out's conjecture and you
might be thinking that well we should
have found a counter example by this
point but on the scale of all numbers 2
to the 68 is nothing I mean the paa
conjecture proposed in 1919 by George
paa asserted that the majority of
natural numbers up to any given number
have an odd number of prime factors the
conjecture was eventually proven false
by C Brian Hassel grve in 1958 when he
identified a counter example what's
remarkable is the value of this counter
example was 1. 1845 * 10 361 that's some
10 340 times bigger than all the numbers
checked for 3x
+1 one way to think about 3x+1 is as
though it's a simple program run on a
touring machine machine the seed number
is the input to the machine so in this
picture 2 the 68 is simply an input tape
68 squares long you can think of them as
a string of ones and zeros or black and
white squares saying that the machine
has transformed every input up to this
68 Square tape down to one should not
give you a lot of confidence that it
will do so for all inputs in fact it's
fairly simple to calculate a number that
shows any arbitrary Behavior you like so
long as it is finite if you want a
number that increases by 3 over two five
consecutive times you can calculate that
number if you want a number that climbs
by 3 over2 10 times in a row or 100
times or a thousand times you can easily
calculate those numbers but after the
finite section you specify you have no
more control and every number that has
ever been tested always falls to one if
there is a counter example it's
virtually impossible that someone's
going to guess it and the space of all
possibilities is too big to search
exhaustively by Brute Force 2 to the
Thousand is not a searchable space so if
we're going to find it we have to find
it by some intelligent process and not
by guess and check I had been on Team
3x+1 for 20 years and then this point of
view I I I I realized that like what do
we really know do we what what it's very
hard to prove a theorem that's
false and so could it be that everyone's
struggling to prove this thing because
it's not actually true and 2 to the 60
is not a lot of evidence and even the
statistical version is maybe true and
not evidence for the non-existence of a
Divergent trajectory somewhere in the 3x
+1 sequence of course there is another
option and that is that we'll never know
that the problem is undecidable in 1987
John Conway created a generalization of
of 3x + 1 it was a mathematical machine
that he called fract Tran and he was
able to show this machine is touring
complete which means it can do anything
a modern computer can do but it also
means that it's subject to the halting
problem a chance that the machine never
stops running and so doesn't give you an
output and this does not prove that 3x+
1 is also subject to the halting problem
but it is possible that given what we
know we may never prove the call out's
Con Ure true or false you're going to be
taught in school that we know a bunch of
stuff and there it's it's a lie they're
all lies here's this stupid little
problem come on really we can't solve
this really you know it just
shows math is hard if anything it shows
that all of the things that we can solve
are Miracles we have no right to have
solutions to all these other problems
for my whole life I've thought of
numbers as these incredibly regular
things full of patterns Symmetry and
repetition but what I'm realizing only
now is just how peculiar numbers really
are you can see this most clearly in the
coral representation from a simple
mathematical operation comes something
intricate organic looking and thus far
intractable to us do all numbers connect
to this structure or is there some
unique filament a spindly little thread
that doesn't connect to any of this that
runs off to Infinity and why is it so
hard to tell I think that's why Paul
Erdos said mathematics is not yet ripe
enough for such
[Music]
questions what I love about 3x+1 is it's
a problem almost anyone can understand
and play around with and actually trying
to figure things out for yourself is the
best way to learn which is why I
subscribe to brilliant the sponsor of
this video now recently brilliant have
upped their their interactivity for
example here is a great lesson on the
Pythagorean theorem so you don't just
remember the formula but you really
understand what it means now brilliant
is a website and an app designed to get
you thinking deeply by engaging you in
problem solving it's one thing to read
through a textbook and think that you
get it but it's quite another to play
with interactives and actually test
yourself as you go and Brilliant curates
the experiences so they get more and
more challenging over time there's
always a helpful tip or explanation that
takes your understanding to the next
level I'd highly recommend their course
on mathematical fundamentals which now
has even more interactivity and it has
topics that are relevant to all areas of
stem and algorithm fundamentals for
anyone interested in coding it's great
to have a resource like this to inspire
you to learn something new every single
day I try to start off my day by
challenging myself with brilliant and so
if you'd like to join me and a community
of 8 million other Learners go to
brilliant.org
veritasium I will put that link down in
the description so I want to thank
brilliant for sponsoring this video and
I want to thank you for watching