Authentic Personal message at 11:13:34 on Wed Jun 12 1991
From: Richard Duffy <zeno> on MADMAN.MIT.EDU
To: mosquito@ATHENA.MIT.EDU
Hello.  Thanks again for giving us access to the saddle-stapler.
At some point you were talking about mathematical patterns that take
quite a long time before they break down.  Here's a really bizarre
example that anyone (like a high-school student) could comprehend:
        Consider sequences defined by s(1) = a; s(2) = b;
    s(n) = least integer such that s(n)/s(n-1) > s(n-1)/s(n-2).
    In other words, s(n) = 1+floor(s(n-1)^2/s(n-2)) for n >= 3.  These
    sequences are similar in some ways to the classically-studied Pisot
    sequences.  For example, if a = 1, b = 2, then we get the odd-indexed
    Fibonacci numbers.
        If we let a = 8, b = 55 in the definition above, then the resulting
    sequence s(n) appears to satisfy the following linear recurrence
    of order 4:
            s(n) = 6s(n-1) + 7s(n-2) - 5s(n-3) - 6s(n-4)
    Indeed, it does satisfy this linear recurrence for the
    first 11,056 terms.  However, it fails at the 11,057th term!
    And s(11057) is a 9270 digit number.
I read this on sci.math, and decided to verify it for myself.  I started
a "dc" program going to calculate the terms, and it was pretty fast for a
while, but starting getting too slow.  So I decided to let a Scheme
computation do it -- its bignum routines are really very impressive --
and after running for 3 days it finished.  The result is true.
Authentic Personal message at 11:16:24 on Wed Jun 12 1991
From: Richard Duffy <zeno> on MADMAN.MIT.EDU
To: mosquito@ATHENA.MIT.EDU
And at n=11057 the recurrence is only off by 1.  Really strange.
athena% zwrite zeno
Type your message now.  End with control-D or a dot on a line by itself.
Does it diverge from that or does it stick close to the formula except for
some flukes?
.
Authentic Personal message at 11:18:47 on Wed Jun 12 1991
From: Richard Duffy <zeno> on MADMAN.MIT.EDU
To: mosquito@ATHENA.MIT.EDU
I didn't check, but I think it gets steadily worse with  n.
The poster claimed:  The reason for this coincidence depends on a 
remarkable fact about the absolute values of the roots of the polynomial
x^4 - 6x^3 - 7x^2 + 5x + 6.  But I never asked him about it.
Authentic Personal message at 11:46:15 on Wed Jun 12 1991
From: Richard Duffy <zeno> on MADMAN.MIT.EDU
To: mosquito@ATHENA.MIT.EDU
Right.  If you should want the exact terms for  n=11053-11057, I
still have them.