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one two four eight sixteen what comes
next
exactly
Answers 31 now I'll explain that soon
but before that I want to go over
something known as the coal lots
conjecture which is an unsolved problem
in mathematics that honestly drives me
crazy not because I could solve it or
anything but it's something anyone can
understand and just seems like someone
should have solved it by now but
apparently it's a very difficult problem
and it goes like this
you pick a positive integer if it's even
you divide it by two and if it's odd you
multiply by three and add one you just
keep doing this with each new number so
let's start at ten it's even so we
divide by two and get five then we apply
the rule again five is odd so we
multiply by three and add one to get 16
16 is even so we divide by two eight is
even so again divided by 2 then again
and one more time now we're at one which
is odd so we now multiply by three and
add one which gets us four then divide
by two divide again and we get stuck in
this loop so now let's try starting at
24 using the same rules we then get 12 6
3 10 5 16 8 4 2 1 and that cycle just
repeats no one knows whether this will
happen for every single number but so
far the sequence has reached 1 for every
single number tested and we've tried
things past 10 to the 18th so this is a
question that I like to bring up to
students because it's something anyone
can understand but when I tell people
how it works and how it's unsolved the
common response I get is well can't we
just assume it's true if we've tried
that many numbers and it works for all
of them and the answer is definitely no
I mean just tell that to this integral
here the answer to this is PI over 2 but
if we had another term of basically the
same thing with 201 instead of 101 the
answer is still PI over 2 if we had
another term with 301 then we still get
PI over
- it's not until you repeat the sequence
about ten to the forty three times that
the pattern is finally broken once you
pass seven point four times ten to the
forty three you will never get PI over
two again sorry that's just the weirdest
thing to me like ten to the 43rd if I
were to stack ten to the forty three
pieces of paper right here and got in a
rocket that could go the speed of light
the time it would take to reach the top
of the stack would be over a trillion
times longer then how long the universe
has existed I just really wanted to put
that number in perspective moving on
let's take a circle and put a single dot
on it kind of a stupid question here but
how many regions did this divide the
middle into well just one it's just the
entire inside but let's add another dot
and connect them with a line
now how many regions do we have
obviously two this time and now let's
add a third dot and connect everything
yes we're making sure every single dot
is connected so now we have four regions
if we add another dot we get eight
regions and then add a fifth dot and it
will get us sixteen regions you think
the next dot would make thirty-two
regions right well in fact adding a
sixth at leaves us with thirty-one
regions and thus the pattern is broken
although this looked like it was going
to be a two to the n equation it is in
fact this plug in the number of dots you
have on the circle in for N and it will
spit out the amount of regions that the
circles divide it into and you'll see if
you were to plug in one two three four
or five then you get the same answers is
just an exponential function with those
same inputs which is probably what you
thought the equation was going to be or
here's the start of another integral
pattern that doesn't break quite as far
down the line as this other one did but
the air is interesting this integral
also equals PI over two if we add
another term with sine of X over five
over x over five it still equals PI over
two and the same with x over seven if we
continue the pattern it fails at x over
fifteen in which the answer is PI over
two minus two point three one times ten
to the minus eleven so the pattern
breaks but by such as
small amount and for anyone wants the
proofs to these I'll provide links below
because they are pretty involved but
hopefully you're seeing the moral of
this story
another example has to do with the
greatest common divisor of this number
and this number where n is any positive
integer if you were to try a bunch of
values for n it would seem like the
greatest common divisor would always be
1 when n is 1 you get 10 and this bigger
number and the only thing that goes into
both of them evenly is 1 for N equals 2
you'd see the same thing in fact if you
made a program that would test every
integer starting at 1 going up and let
it run for the rest of your life you
never come across anything where the
greatest common divisor is not one
that's because the first counter example
is this these numbers are seeing don't
necessarily have any significant meaning
beyond these patterns but it's still
crazy how you can't just assume anything
really I left this one for later because
it's very well known but if you look at
the polynomial N squared plus n plus 41
it seems like any integer value for n
will yield a prime number if you plug in
1 you get 40 30 which is prime plug in 2
and you get 47 which is also prime you
plug in 30 and you get 971 also Prime so
it seems like we maybe found a simple
polynomial to locate Prime's but if you
plug in 40 you get 16 81 which is not
prime it's actually 41 squared and you
don't need a calculator to see that if
you plug in 41 for n it's definitely not
prime since it's divisible by well 41 if
you guys have seen the movie the man who
knew infinity which I highly recommend
if you haven't you remember that the
main character ram anujan becomes
irritated they's forced to learn proofs
this is saying that he didn't have
formal training with and being extremely
gifted he was confident in the theorems
that he found without proving them but
there's one scene where his adviser
finds an errand his work that comes as a
result of him not providing a proof so
hopefully you're seeing that no matter
who you are mathematicians require
proofs before they'll believe anything
and I hope this video is kind of showing
you why that is this really applies to
pure mathematics more than anything but
I still thought the patterns we saw here
we're really interesting and that anyone
can understand them but it highlights
that you can't
any assumptions too early and before I
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