[seqfan] Re: Proving a number to be prime using the Tower of Hanoi.
Robert Munafo
mrob27 at gmail.com
Sat Oct 18 08:32:07 CEST 2014
On Tue, 14 Oct 2014 11:17:23, Daniel Joyce <hlauk.h.bogart at gmail.com> wrote:
> The tower of Hanoi and the sum of its pieces in all the moves it takes to
move to a different peg is related to the primes and a sequence in OEIS.
> ...
Did you try 341? :-)
Looking at A000295, note the direct formula a(n)=2^n - n - 1, and the
recurrence formula a(1)=0, a(n)=2*a(n-1)+n-1, n>1.
Now consider a given "possible prime" p. Daniel has suggested looking at
the continued fraction (c.f.) expansion of the fraction a(p+1)/a(p). For
example, is p is 5, we are considering the c.f. expansion of a(6)/a(5),
which is 57/26.
For any given p, a(p) will be 2^p-p-1 (example: 32-5-1 = 26), and the next
term a(p+1) is 2*a(p) + (p+1)-1, or more simply 2*a(p)+p. (example:
2*26+6-1 = 52+5 = 57)
In general, we want to find the terms of the c.f. expansion of
[2*(2^p-p-1)+p]/(2^p-p-1). You can see right away that this is 2 +
p/(2^p-p-1). So the first term of the c.f. is 2. Continuing, we have to
turn it into
2 + 1/[(2^p-p-1)/p]
and consider what happens with the next term.
If p is, in fact, prime, then by Fermat's Little Theorem, 2^p-2 is a
multiple of p. If we had "(2^p - p - 2)/p" it would be an integer and our
c.f. would be done. Instead we are too big by 1, so there will be 1/p left
over. In other words, (2^p - p - 1)/p = k + 1/p for some integer k. That
1/p results in one more c.f. term and then the c.f. terminates: 2 + 1/(k +
1/(p + 0)).
If p is composite, then *usually* 2^p-2 is not a multiple of p, and the
c.f. ends up having more terms. But not always... there are composites for
which 2^p-2 *is* a multiple of p. These are called "*Fermat pseudoprimes*"
to base 2.
I'll use Maxima, because it has a nice c.f. function and you don't have to
set a precision limit. Here is what happens when p=5 (n is numerator, d is
denominator)
(%i8) p:5;
(%o8) 5
(%i9) d:2^p-p-1;
(%o9) 26
(%i10) n:2*d+p;
(%o10) 57
(%i11) cf(n/d);
(%o11) [2, 5, 5]
So far, so good. Now we'll let p be the composite number 9, and then the
known prime 37:
(%i1) p:9;
(%o1) 9
(%i2) d:2^p-p-1;
(%o2) 502
(%i3) n:2*d+p;
(%o3) 1013
(%i4) cf(n/d);
(%o4) [2, 55, 1, 3, 2]
(%i12) p:37;
(%o12) 37
(%i13) d:2^p-p-1;
(%o13) 137438953434
(%i14) n:2*d+p;
(%o14) 274877906905
(%i15) cf(n/d);
(%o15) [2, 3714566309, 37]
Everything seems fine so far. But here's what happens with 341:
(%i16) p:341;
(%o16) 341
(%i17) d:2^p-p-1;
(%o17)
44794894843556084211148845611368885562432909944692990697999782019275837\
42360321890761754986543214231210
(%i18) n:2*d+p;
(%o18)
89589789687112168422297691222737771124865819889385981395999564038551674\
84720643781523509973086428462761
(%i19) cf(n/d);
(%o19) [2,
1313633279869679888889995472474160866933516420665483598181811789421\
5788100763407304286671514789484549, 341]
(%i20) factor(341);
(%o20) 11 31
In line %o19 we see that the c.f. expansion terminates after just 3 terms
(the middle one is big), but in line %o20 we see that 341 = 11 * 31.
On the plus side, if you actually manage to complete a Tower of Hanoi with
341 disks, you will have out-lived the Universe, which is quite an
accomplishment, primes or no primes.
--
Robert Munafo -- mrob.com
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