These are called Friedman numbers. It can
be written in a non-trivial way just
using the digits itself together with
any combination of your usual
mathematical signatures; plus, minus, times,
divide; you can even raise things to a
power; and you can use some combination
of brackets if necessary. So 25 is a
Friedman number, ok? Why's 25 a Friedman
number? Because it can be written, in
terms of the two digits that make it up,
as 5 to the power 2. So I've just used
the raising power operator to do that.
121, same sort of idea: two 1s form
11 and I raise it to the power 2 and
I've got 121- (Brady: Immediately I'm seeing)
(this is- could be described as)
(recreational mathematics?) - I- it's the kind
of thing I used to have great fun
playing with, but actually there's some
really neat things you can do with it.
The brackets you've got to be a little
bit careful with, you- you're not allowed
just to simply wrap up one number up in
a bracket. So 125 is 5 to the power of 2
plus 1, so I'm just using the brackets
there to add the 2 and the 1 and make up
3, so 5 cubed is 125. Well if I wanted to
go for one say with 4 digits then 9261
is 21 to the power of 9 minus 6. So
there's actually 72 of these numbers
with 4 or less digits, and there's a-
there's a really nice sort of subset of
them. 14 of these are known as nice
Friedman numbers and a nice Friedman
number is where the order of the digits
remains the same. So a simple one is 127;
127 is made up of minus 1 plus 2 to the
power 7. The order of the digits 1 2 7
is the same as appears in the- on the
right hand side. Whereas if I say took
125, the left hand side is 1 2 5 but the
right hand side is 5 2 1, or 5 1 2.
(Is that not nice?) - Not formally nice.
It's a- for many of us we quite enjoy
125, it's got many nice properties,
but on this occasion it's not a nice
number, not a nice Friedman number anyway.
So you could do- I mean if I want to just
get a bit bigger it's- 343
for example is just made up of 3
plus 4 to the power 3; or I could
get a bit bigger, and 6455
is made up of 6 to the power 4 minus
5, all times 5. So you get the idea
that there are- you can make up various
combinations and in fact we've already
come across a class of these - vampire
numbers - that we've done videos with. So
vampire numbers are made up of these
where you instead of using all the
various combinations we're just using
the multiplication. So- so vampire numbers
are an example, so 1260 is a
vampire number, but it's also a Friedman
but because it's 21 times 60. You
know, you can get a bit bigger. So the-
we've got five digit numbers of which
there are 272 of
them by the way; 10192, which is 101 to
the power 2 minus 9. And if I went to
a 10 digit one: 8326197504,
what's that equal to?
91248
to the power of 5 minus
3, plus naught times 67.
(Is the naught times 67 together? Have they just)
(used the naught there to cheat away- ?) - Well
the naught's in here, and so it's part of it-
but what- it does make it- there is a
an interesting caveat which we have to- I
should have mentioned. You can't have
leading zeros, they're not allowed. For
example, something like
001729 and write that as
1700 plus 29. I'm not
allowed to have the leading zeros in
there in that definition. But there is
some neat one- there's some even more.
(Better than this even?) - It gets better. What's
the smallest number that I can- a
smallest Friedman number that I make
just repeating the same digit?
(These are called repdigits aren't they?) - Repdigits. Repdigits, okay? And
the smallest is one, two, three - these are
all nines - one, two, three, four, five, six,
seven, eight; one, two, three, four, five, six,
seven, eight. And it's made up of 9
plus 9 over 9, to the power of 9 minus 9
over 9, minus 9 over 9.
How about 1s? So now we've got one, two,
three, four, five, six, seven, eight, nine, ten,
eleven. And that is equal, if I've got my
algebra right, 11 minus 1 to the
power of 11, minus 1 times 1, divided
by 11 minus 1 minus 1. So they've got the same
number on the right hand side and- - (Why are you laughing?)
Because it is just cool. But I haven't
finished, there's more. How about if I ask
the following, there's a set of
numbers called your pandigital. - (We have)
(covered pandigital numbers.) - Oh you've done?
Yeah, okay, well this- but you haven't done
pandigital Friedman numbers I bet.
Here's one, right, so a pandigital number is
one that includes all the numbers in
any given base, so if I say
123456789; well I
can write that as a Friedman number
because that's equal to - think I need two
brackets - 86 plus 2 times 7 to the power
5 minus 91 all divided by 3 to the
power 4. So that's one pandigital number. We
can do another one.
987654321, right? And this is 8
times 97 plus 6 over 2 to the power 5
plus 1, again divided by
3 to the power 4. You might say, is it
easy to get a nice pandigital number? And
indeed they're there, but they're not
that easy to get, there is one here:
268435179.
And that is equal to- remember the-
what's nice about it is the order
remains the same, the order of the
numbers remains the same. So it's minus
268 plus 4 to the power of 3
times 5 minus 1 to the power 7
minus 9. And that will give you back
your original number. It turns out, as the
number of digits and the number
increases the likelihood of it being a
Friedman number also increases. If there
are F of n Friedman numbers less than
any given number n then it turns out, and
I don't know the proof of this I haven't been
through it, that as n increases the ratio
of F of n over n tends to 1. In other
words, the likelihood of finding a number
tends to unity. So given a distribution
of numbers, you know of large n, then you
pick a number at random then the
likelihood is that it's it's actually a
Friedman number. - (So so the higher we go)
(in the number line the more dense it becomes with Friedman numbers?)
Yeah, yeah, I think that's right.
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