New sequence??

[John Conway]

On Fri, 2 Apr 2004, Richard Guy wrote:

> Sorry about that!!  I wasn't including n.
> 
> Shift all my members one place.     R.

ALL your members??!!   JHC

 
> On Thu, 1 Apr 2004, Richard Guy wrote:
> 
> > If  n = 26,  we want the least  t_n  so that some
> > subset of  26 + 1, 26 + 2, ..., 26 + t_n  has a
> > square product.  I make this to be
> > 
> >          27* 28 * 30 * 32 * 35     [...]

     I wonder how many sequencers are aware of the fact that
for any sequence  a,b,c,...  of positive integers, the sequence
a, a^b, a^b^c, a^b^c^d, ...  converges to any modulus?

    A remark of Jerrold Grossman's led me to discover this for
myself, and it pleased me very much, because of course it makes
infinite "numbers" like  10^10^10^...  meaningful modulo any number.
For instance, that one is congruent to 38 modulo 47.  

   However, when I mentioned it to Dick Bumby last weekend, he said
that this theorem had been proposed in the Monthly when he was Problems
Editor, and had produced a lot of correspondence, some of which was
from Jerry Grossman, and some of which gave early references, so the
I'm far from being its first author.  Oh well!

   It gives rise to lots of sequences, of which my favorite is that
giving the least positive remainders of  2^3^4^5^...  modulo 2,3,4,5,... .

    JHC