[seqfan] Re: 'Climb to a prime' in other bases

[David Seal]

I can now add base 15 to the list of bases for which the 'Climb To a Prime' conjecture (that all trajectories end in a prime) is known to be false, as (using digits A=10, B=11, C=12, D=13, E=14):

base-15 144D299E1
= 3301896361
= 19 * 73^2 * 32611
= base-15 14 * 4D^2 * 99E1

Best regards,

David


> On 15 June 2017 at 20:35 Neil Sloane <njasloane at gmail.com> wrote:
> 
> 
> David Seal:  Nice work!
> 
> It turns out that the base-2 version is already in the OEIS,
> although without your counterexample, which I have now added.
> 
> I also added some sequences that were missing, so the base-2 problem is now
> described in the following 5 entries:
> A230625 & A287874 for the basic map,
> and, for what you reach when the map is
> iterated, A230626, A230627, A287875.
> 
> There is a slight awkwardness in A287875, because the escape clause, which
> says we write -1 if we don't reach 1 or a prime, is tricky to deal with in
> base 2.  So the escape clause is "a(n) = -1 in decimal if ...", and
> otherwise "a(n) = the prime reached, or 1, written in binary".
> 
> Did you check to see if 255987 is the smallest number that is fixed in base
> 2?  I assume so, but it wasn't clear from your message.
> 
> It would be interesting to see the list of numbers that don't reach 1 or a
> prime.  That is, the numbers that are fixed (and composite), or go to a
> composite fixed point, or go into a loop.  If you have this list, could you
> submit it as a new sequence?
> 
> 
> Best regards
> Neil
> 
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
> 
> 
> On Tue, Jun 13, 2017 at 9:31 AM, David Seal <david.j.seal at gwynmop.com>
> wrote:
> 
> > The recent posts about 13532385396179, http://oeis.org/A080670 and
> > http://oeis.org/A195264 caused me to wonder whether similar numbers exist
> > when you write them in bases other than 10. I've done some computer
> > investigations of the question, so far only using 'brute force' programs
> > (I've got a number of ideas how to improve on them), with the following
> > results:
> >
> > First, there are a couple of very simple examples in bases 6 and 8:
> >
> > base-6 24 = decimal 16 = decimal 2^4 = base-6 2^4
> >
> > base-8 33 = decimal 27 = decimal 3^3 = base-8 3^3
> >
> > More generally, for any prime p and integer n > 0, the 2-digit number with
> > digits p, np is equal to p^np in base b = p^(np-1)-n. That doesn't lead to
> > a solution for the (p,n) = (2,1) combination, since that implies b=1 and so
> > has the problem that the digits are >= the base, but otherwise it works,
> > leading to a doubly-infinite family of bases in which examples exist, the
> > next one of which is 29:
> >
> > base-29 26 = decimal 64 = decimal 2^6 = base-29 2^6
> >
> > Other than that, I have tried all bases from 2 to 20, and found examples
> > in bases 2, 11 and 12:
> >
> > binary 111110011111110011
> > = decimal 255987
> > = decimal 3^3 * 19 * 499
> > = binary 11^11 * 10011 * 111110011
> >
> > base-11 3518
> > = decimal 4617
> > = decimal 3^5 * 19
> > = base-11 3^5 * 18
> >
> > base-12 15287
> > = decimal 29767
> > = decimal 17^2 * 103
> > = base-12 15^2 * 87
> >
> > In binary, I've also found a couple of 'amicable pairs', in which the
> > operation of writing down the factorisation and then concatenating the
> > primes and exponents on each number in the pair produces the other one. The
> > first is:
> >
> > binary 1111101111
> > = decimal 1007
> > = decimal 19 * 53
> > = binary 10011 * 110101
> >
> > binary 10011110101
> > = decimal 1269
> > = decimal 3^3 * 47
> > = binary 11^11 * 101111
> >
> > and the second is:
> >
> > binary 10111011111
> > = decimal 1503
> > = decimal 3^2 * 167
> > = binary 11^10 * 10100111
> >
> > binary 111010100111
> > = decimal 3751
> > = decimal 11^2 * 31
> > = binary 1011^10 * 11111
> >
> > And the first of those two happens to work in base 4 as well:
> >
> > base-4 33233
> > = decimal 1007
> > = decimal 19 * 53
> > = base-4 103 * 311
> >
> > base-4 103311
> > = decimal 1269
> > = decimal 3^3 * 47
> > = base-4 3^3 * 233
> >
> > So the 'climb to a prime' conjecture is false in bases 2, 4, 6, 8, 10, 11,
> > 12, 29 and the various higher bases of the form p^(np-1)-n with p prime, n
> > >= 2.
> >
> > There are also various negative results - these are obviously dependant on
> > there being no as-yet-undetected bugs in the programs I've written. They
> > search for length-L "loops" in iterative application of the operation,
> > where a number produces itself after L applications of the operation (so
> > the 'amicable pairs' are length-2 loops and the other examples above are
> > length-1 loops). More specifically, they search for non-prime maximal
> > elements of loops, which are obviously in 1-to-1 correspondence with loops
> > other than the length-1 loops that contain a prime number only. The results
> > are:
> >
> > Bases 2-6: No non-prime maximal elements of loops found other than those
> > listed above (1269, 3751 and 255987 in base 2; 1269 in base 4; 16 in base
> > 6) up to 6.5*10^8. (Programs are still running, so that limit will probably
> > increase over time.)
> >
> > Bases 7-20: No non-prime maximal elements of loops found other than those
> > listed above (27 in base 8; 4617 in base 11; 29767 in base 12) up to 7^10,
> > 8^10, 9^9, 10^9, 11^8, 12^8, 13^8, 14^7, 15^7, 16^7, 17^7, 18^7, 19^7, 20^6
> > respectively. (These results were obtained by an earlier, memory-bound
> > version of the program, which I am no longer running.)
> >
> > There's obviously the beginnings of an integer sequence here, defined as
> > the smallest maximal element of a sequence generated by the 'climb to a
> > prime' operation in base N that doesn't actually climb to a prime, or -1 if
> > no such sequence has a maximal element. But 1269, ?, 1269, ?, 16, ?, 27, ?,
> > ?(<=13532385396179), 4617, 29767, ... is probably a bit too little
> > knowledge to register an OEIS sequence on!
> >
> > David Seal
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
> 
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