NetNews Usenet Archive 1992 #18/NN_1992_18.iso/spool/sci/math/10389

Text File | 1992-08-18 | 3.2 KB | 109 lines

Newsgroups: sci.math
Path: sparky!uunet!usc!sdd.hp.com!mips!darwin.sura.net!jvnc.net!nuscc!bhonsle!bhonsle
From: [e-mail] (Shailendra K Bhonsle)
Subject: Re: u(v^n)w prime puzzle
Message-ID: <[e-mail]>
Sender: bhonsle@bhonsle (Shailendra K Bhonsle)
Organization: Institute of Systems Science, NUS, Singapore
References: <[e-mail]> <[e-mail]>
Date: Wed, 19 Aug 1992 02:41:06 GMT
Lines: 97

In article <[e-mail]>, [e-mail] (Kelly Roach) writes:
|>
|>
|>
|>      Prove or disprove:  There are three non-empty
|>      strings of digits u,v,w such that all the
|>      numbers in
|>           L = {u(v^n)w | n is a natural number}
|>             = {uw, uvw, uvvw, uvvvw, uvvvvw, ...}
|>      are prime numbers.
|>
|>
|>      Time to say a few more words about my u(v^n)w prime
|> puzzle which I posted yesterday.  I am definitely not looking
|> for any solutions involving u=v="0".  Ordinary syntax only
|> please.  No leading zeros in u.
|>      Some interesting patterns:
|>
|>      u="3",v="3",w="1"
|>      31, 331, 3331, 33331, 333331, 3333331, 33333331
|>
|>      u="1",v="36",w="1"
|>      11, 1361, 136361, 13636361, 1363636361, 136363636361
|>
|>      u="17",v="57",w="09"
|>      1709, 175709, 17575709, 1757575709, 175757575709,
|>      17575757575709, 1757575757575709, 175757575757575709
|>
|> Below each line giving u,v,w values appear a lot of prime
|> numbers.  These patterns do eventually fail:
|>
|>      333333331 = 17*19607843
|>      13636363636361 = 17*1321*5693*106661
|>      17575757575757575709 = 232433*75616446785773
|>
|> The question is, are there any {u,v,w} examples which do
|> not fail?  That always give prime numbers?
|>      I know the solution to this puzzle.  I think the
|> solution can be understood fairly easily by anyone that
|> has had a first course in number theory.  See if you
|> can discover it.
|>
|>
|>
|>
|>
|>

--

Hi,
I think the answer is no, it is not possible to have such a set of u v and w.
Since I am not a hardcore mathematician (never done anything in number theory
formally), let me first give me the intuition and then the proof:

1. Intuition: If it were possible then it will not be such a big deal to
discover the biggest prime number etc. (just keep on repeating "v") provided
one is able to find u,v, & w.

2. proof:

The approach is to take a integer x (let's say prime integer) and prove that by repeating
v one gets a string u(v^m)w (for positive integer m) which is divisible by x.

Let's take x=9 (because it has nice properties for base 10 numbers).

then for a number "uv"(ie. 10u+v)
     uv=u+v (mod 9)


In our problem definition:

let u+w=m(mod 9)

Now it is possible to find a n for given v such that

                     v^n==n.v== 9-m (mod 9) ==> simple number theory


Hence for this n u(v^n)w= u+w+n.v==0(mod 9)



So for any such u,v,w there is at least one number which is divisible by 9.


So by contradiction we prove that it is not possible to have such triples.


-----------------------------

Is this proof satisfactory??

Shailendra
([e-mail])
Inst. of Systems Science, Singapore