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How to pronounce 307 in English?

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a huge thanks to my patreon supporters
for making this episode possible
[Music]
economic mutations way come back to now
I know for the longest time now since
the start of the channel basically
people have wanted me to do more
competitive mathematics problems and I
fought yet why the hell not
it's going to get me some new viewers I
suppose why not we are going to die for
them
the first one is going to be an aim 1
2015 problem problem number 3 and it's a
number theory problem basically what's
the smallest prime P with 16 P plus 1
being the cube of a positive integer
meaning we can rewrite this at first as
16 P plus 1 is equal to some I don't
know R to the 3rd power
it doesn't quite matter what you call
this thing R is all of the positive
integers so n now we just need to take
two cases at first so so whenever I
present a problem like this I'm going to
show you how I solved it the first time
around it's it's not like I'm optimized
in in any way probably it's just how I
solved it and and I don't know that's
that's mostly the best kind of solving a
problem how you did it on your own so at
first there are two cases for prime
numbers but or even Prime's the case
with the even Prime's it's really easy
to check ok so the only even prime is 2
so what happens if P is equal to 2 well
then we have 16 times 2 is 32 plus 1
it's nothing about 33 being equal to
some R cubed with 33 it's not a perfect
cube it's it's not there's not a perfect
cube you could find that gives you 33
it's just not something that happens all
right so with that case out of the way
we know that when Pete must be aught so
that's the first thing I did then well
mmm how can we proceed from this point
onwards very cool thing is get something
called this geometric progression if you
take a look at
geometric progression this is just after
form K is bounded between 1 and n in
some way X to the K power and this is
equal to well X to the n plus 1 power in
this case minus 1 over X minus 1 you
might have seen this before if you let n
go to infinity then this is just a
geometric series meaning of wall overall
if we choose n to be equal to 2 then
that means that we have X to the 3rd
power minus 1 being equal to if we
multiply both sides by X minus 1 we
don't want it to be equal to 0 for now
we don't know what the roots are going
to be of this polynomial then we are
going to end up with in expression for X
to the 3rd power minus 1 in some way how
can we get X to the third power minus 1
well in our case it's our and well why
not just subtract 1 on both sides
leaving us with in the first step R to
the third power minus 1 being equal to
16 P well now we exactly have a
geometric progression here meaning over
all 16 P is thus nothing but okay on the
one hand we are going to have our minus
1 as a factor in some way so we are just
going to affect a third degree
polynomial pretty easily using this and
then what are we going to have as the
other factor it's just a sum that runs
from 1 to well n in our case has been 2
so this is nothing but R squared plus R
plus 1 and here it's as easy as it is
now we can go through a few more cases
overall so at first we could divide both
sides by 16 so why not do that
meaning overall P R our prime that we
are looking for the smallest prime over
all it's nothing but our minus 1 over 16
times R squared plus R plus 1 receiving
from this point how could you do this
well we could take for the parity for
example parity means are we going to get
an odd or an even number if we plug some
integer here some positive integer so
let us check it first what happens if
for example are aware I don't know even
for us
so first case case one hour even a deaf
equivalent to saying that our our is
nothing but 2 times n where n this
element of the positive integers in this
case just because we wanted R to be a
positive integer then this means dead
let's just go through the kind of parity
and argumentation here we are going to
get something okay an even number minus
1 for example 2 minus 1 is going to give
us in odd number because this is 1 so we
are going to have odd over even in some
way okay then our pier is odd over even
times hmm okay even number times even
number two times two is four so this is
even so this gives us even plus even
this is 2 plus 2 4 examples also even
plus 1 is going to give us an odd number
so this is odd over even times odd in
all number times not number 5 times 5 is
25
it's going to give us an odd number so
the parity of the swing is odd over even
overall but odd over even is never going
to be in the natural numbers for example
the positive integers I have proven this
before I believe in my how the
mathematicians prove things linked with
it or dare to talk the description
probably just as an easy example let's
take 3 over 2 it's it's 1.5 it just
doesn't work because this thing is an
even number plus 1 over even gives us 2
over 2 plus 1/2 in this case really
doesn't work out okay this is not
element of n meaning overall our our
cannot be even in this case meaning the
only thing that could happen is that our
our thought case number 2
our spot and that's equivalent to saying
that our is of the form 2 and plus 1 or
2 and minus one really doesn't matter ok
it's the same thing where n is element
of n if you do 2 n minus 1 you need to
place some more restrictions on your end
that you're going to have here because
positive integers implies it's without
zero okay just
and without negative 1 in this case okay
just as a little side note so if R is
thought what is the parity of this thing
let us go through this at first just to
verify if everything works out P must
also be odd then ok it's it needs to be
an odd number so P is does in parity R
minus 1 is 2 n plus 1 minus 1 is going
to give us overall even number of an
even number times
well what times odd is going to be odd
we had this before or plus up five plus
five is even even plus 1 is going to be
odd
it's subway so we are going to have even
over even times odd even times ought to
times 5 is going to be even yet again so
this is even over even if this does work
out 4 over 2 is going to be a well - so
this works out could be element of
natural numbers there's no contradiction
here it could so now let us write out
what we can gather for our P here by
replacing our in some way with this 2 n
plus 1 ik expression and I'm just going
to do this for the first part because
well it's it's just way easier then then
our P R is equal to well 2 n plus 1
minus 1 over 16 okay plus 1 minus 1 is
going to just cancel out to 0 ok and
then we have 2 n over 16 well it's
common factor of 2 on the numerator and
denominator this is just going to give
us n over 8 overall I hope you could
follow everything I set here then we are
going to leave a square how it is and
all the other ours here so R squared
plus
plus one now
we trust me to plug in some ends with
with n at our hands we also have
definition for R or value and then we
are going to see if we can find a prime
number what happens for example if we
choose n to be less than 8 for example
what were to happen well then we would
have for example for n being equal to 1
let us just go with n being equal to 1
in the type instead R is equal to 3 in
our case and this means that P is thus
ok this is 1 8 times and then we just
have some some other factor here so this
gives us free squared plus 3 is going to
be 12 plus 1 is going to be 13 13 over 8
it's not an element of n and this is
going to happen if n is any less than 8
what were to happen so with each in
every case where 8 n is not or is less
than 8 in some way we are going to get a
fraction out just because of our and
being less than what we have done here
it just is what is this okay just going
to happen then what happens if our n is
exactly equal to 8 if n is equal to 8
then P is obviously okay 8 over 8 is
going to cancer all is going to be 1 if
n is equal to 8 then we are going to get
17 for our R so this means this is 17
squared plus 17 plus 1 this is going to
give us 17 times 17 is 289 plus 17 plus
1 this is 292 90 plus 17 is going to
give us 307 now we need to check if it's
a prime number and if it's a prime
number then it's the smallest one we
could actually have for this problem and
then we are basically done for this I'm
going to toss make use of the square
root test meaning we are going to take
the floor of the square root of 307 and
then we see what
biggest factor basically is that we can
multiply something by such that we could
compose 307 out of prime numbers
basically so square root of 307 is so 17
times 17 is 289 so this is probably the
closest we could get to there so there's
probably 17.3 I don't know what it is
but if we take the floor of this thing
it's just going to be 17 now we just
need to check all the prime numbers up
and 217 if they divide our 307 and then
we're done so 2 obviously doesn't divide
307 because it's an odd number 3 does
also not divide 307 because 3 divides
306 so it doesn't work out for 5 no
doesn't work out because well we don't
have a seer or 5 at the end what else is
4 7 7 doesn't work out because 350 minus
42 is going to give us 308 so just
doesn't work out for the number 7 what
about the number 11 11 doesn't work out
because the alternating sum is a 3-0
plus 7 is going to be 10 it's not equal
to 0 then we have 14 that's also
something that could work out this is
going to give us 260 260 and then 5
times this is going to give us 300 no
doesn't work out and this has been the
less prime number before 17 and 17
doesn't divide the string because well
we have plus 1 here ok if it would have
another 17 here then it would be
divisible by 17 but it doesn't meaning
307 smallest prime I hope you could for
everything I did here so I just went in
my head through all the prime factors
they could be less or equal to 17 and
none of those divided 307 and does is
why it's a prime number and it's also
the smallest one here I think that's
watching if you enjoyed this video
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want support end that Mobile Suit as I
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tell me so hello kitty kitty kitty cat
is katie katie is nice to have you here
I would love to draw some your viewers
in and I believe that posting some math
competition problems is going to do the
trick
over time so send me some math
competition problems if you have some at
your hands and I went to next video I
will show you guys flambo day yeah
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a huge thanks to my patreon supporters

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How to pronounce 307 in English?

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