On 9-8-2012 12:15, Invisible wrote:
> Warp's challenge on prime numbers got me thinking...
>
> The product of two integers is always an integer. However, the quotient
> is not always an integer. When it is, we say that the integers are
> "divisible". This idea leads directly to the concepts of "factors" and
> "divisors", and eventually to the "prime numbers". Famously, every
> natural number is uniquely expressible as a product of prime numbers.
>
> Addition has no such properly. For any given integer, there are an
> unlimited number of integer sums which produce it. And that's because
> unlike a quotient, the difference between two integers /is/ always an
> integer.
>
> Looking back at multiplication again, suppose we move from the integers
> to the rationals. Now suddenly the concept of divisibility goes away.
> /Every/ rational is "divisible" by /every/ other rational. (With the
> exception of zero, anyway.) All the interesting structure has
> disappeared. (Unless we continue to treat the integers as a "special"
> subset of the rationals...)
>
> To summarise: Multiplication has an interesting structure in the
> integers, but that structure goes away when we consider a larger set
> (the rationals).
>
> Addition does not have an interesting structure on the integers.
> Question: Is there some subset of the integers which /would/ have a
> similar, interesting structure?
>
>
>
> It turns out you /can/ actually do this. And since it can be done,
> unsurprisingly various mathematicians have done it. There's even a term
> for it: It's called a numerical monoid.
>
> According to Jumanji, "In the jungle you must wait, 'till the dice read
> five or eight."
>
> So, what happens if we start with the only integers being 5 and 8. What
> other integers can we produce by adding these?
>
>    5 + 5 = 10
>    5 + 8 = 13
>    5 + 5 + 5 = 15
>    8 + 8 = 16
>    5 + 5 + 8 = 18
>    5 + 5 + 5 + 5 = 20
>    5 + 8 + 8 = 21
>    5 + 5 + 5 + 8 = 23
>    8 + 8 + 8 = 24
>    ...
>
> In particular, there is no way of making 27, but you can make 28 just
> fine (5+5+5+5+8). Actually,
>
>    28 = 5+5+5+5+8
>    29 = 5+8+8+8
>    30 = 5+5+5+5+5+5
>    31 = 5+5+5+8+8
>    32 = 8+8+8+8
>    33 = 5+5+5+5+5+8
>    34 = 5+5+8+8+8
>
> According to my simulations, you can make /every/ number above 27. So 27
> is the highest number that you cannot make.
>
> When you get to 40, something interesting happens. For 5*8 = 40, which
> means that we have
>
>    5*8 = 5+5+5+5+5+5+5+5 = 40
>    8*5 = 8+8+8+8+8       = 40
>
> So this number can be made in /two/ different ways.
>
> When we come to 45, the same happens again, since 45 = 40+5, and 40
> itself can be made two different ways. So you can take both of those
> ways and append +5 to make 45 in two different ways.
>
> The same happens again with 48 = 40+8.
>
> It happens yet again with 50 = 40+10. 10 can only be made one way, but
> 40 has two ways.
>
> It happens yet again with 13, since 53 = 40+13, and 13 is makable.
>
> In fact, above 27, /every/ number can be made at least one way, 40 is
> the first number that can be made /two/ ways, and every number which can
> be made by adding a makable number to 40 also has two ways.

Does that make 67 the highest number you can make in only one way?


-- 
Women are the canaries of science. When they are underrepresented
it is a strong indication that non-scientific factors play a role
and the concentration of incorruptible scientists is also too low

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