Re: prime numbers and African artifact

Martin Bright (martin@melon.wdg.uk.ibm.com)
14 Jul 1995 08:37:49 GMT

Peter Seebach (seebs@solutions.solon.com) wrote:
|> In article <3u0rjh$g2g@zen.hursley.ibm.com>,
|> Martin Bright <mbright@vnet.ibm.com> wrote:
|> >Peter Seebach (seebs@solutions.solon.com) wrote:
|> >|> > What's the easy way to show that 254365465431652436514232 is not
|> >|> >prime again?

|> |> Simple!

|> >|> 2 + 5 + 4 + 3 + 6 + 5 + 4 + 6 + 5 + 4 + 3 + 1 +
|> >|> 6 + 5 + 2 + 4 + 3 + 6 + 5 + 1 + 4 + 2 + 3 + 2 = 91

|> >|> 9 + 1 = 10.

|> >|> 1 + 0 = 1.

|> >|> 1 is not a multiple of three, so the whole number is not a multiple of
|> >|> three, so it must be prime!

|> >Also the first two digits aren't divisible by 4, so the whole number isn't
|> >divisible by 4. This proves again that it is prime.

|> Actually, this is incorrect. If the *last* two digits are divisible by 4,
|> the whole number is divisible by 4. However, since they aren't (they're 3
|> and 2), the number isn't divisible by four. (The way this works is, since
|> it's not a multiple of four or a multiple of three, it's not a multiple of
|> (4 * 3), so it's not a multiple of 12 - like 13, which is also prime.

OK, I'm starting to get the hang of this. If every digit of a number is
divisible by 2, then the number itself is divisible by 2. [Note that the
preceding statement is in fact true]. However, there are definitely a few
1s, 3s and 5s in there, so that number is not divisible by 2.

Hang on, I think I can finally wrap this up.

(a) 254365465431652436514232 is not divisible by 2 (see above).

(b) Since the first number is neither a 5 nor a 0, the number is
not divisible by 5.

(c) By repeatedly adding the digits together as above, we get the number 10.
Now any number which doesn't divide into 10 clearly doesn't divide into
the whole number. 10 is divisible only by 5 and 2, which we have already
shown don't divide the number.

So 254365465431652436514232 is most definitely prime, not to mention square
(yes, a member of the small set of square primes) and also a Fibonacci number.

Thank you.

Martin

--



Martin Bright, Pre-University Employee    |  mbright@vnet.ibm.com

Warwick Development Group, IBM UK         |  9BRIGHM at NHBVM9