From k.buzzard@ic.ac.uk Tue Feb 2 22:22:08 1999
Date: Tue, 2 Feb 1999 13:29:40 -0500
From: Kevin Buzzard
To: NMBRTHRY@LISTSERV.NODAK.EDU
Subject: A fairly-innocent looking equation
My boss (a group theorist) has run into an equation that he'd like
to "solve": it's
x^2+7=8p^n
where x is an integer, n>=0 is an integer, and p>=2 is prime.
My subject area isn't really equations of this sort, but I had a bash
anyway because it looked like fun. Initially he wanted to know the set
of solutions. But even for n=1 the question basically reduces
to something like "which primes are of the form f(m)" where f is a polynomial
of degree 2. So I guess one could happily conjecture that there
were infinitely many solutions and also be pretty sure that it was
not currently possible to prove this.
Having been presented with this obstruction, my boss said that
it would suffice for him if it were possible to show that
there was an integer N such that there were no solutions
to the equation with n>N. I don't even know whether it is
reasonable to conjecture that such an N exists, let alone
to prove that one does. But it struck me that there could
be people reading this mailing list that might know about this kind of
thing, or who would know something in the literature...
OK, that was the question. Now here's what I know.
For n=0 we can solve by hand. For n=1 the question is to find the
primes of the form (x^2+7)/8 and there are almost certainly infinitely
many.
For n=2 it gets a little more interesting. The equation becomes
one of "Pell type", and if we forget for a minute that p has to be
prime then we can solve for x and p in integers using standard techniques
and the question has become "here's a degree 2 recurrence relation,
how many primes occur in it?". Well I don't even know whether it's
conjectured that there's infinitely many primes in, say, the
Fibonacci series, so here I'm a bit stuck. I computed the first
few thousand terms of the (two) series in question and found some
_very_ impressive-looking solutions where p had about 600
digits (and I guess also was only technically "probably prime").
So maybe one could conjecture that there were infinitely many
solutions with n=2. But this doesn't bother us because we're
only looking for an N.
For n=2t, t>1, we can use the same analysis above and the question
becomes "here's a series generated by a recurrence relation, how
many t-th powers of primes are in it?". I guess there's only finitely many
squares in the Fibonacci sequences, but there are other sequences
(e.g. 1,2,3,4,5,...) defined by a second-order recurrence relation
that have infinitely many squares of primes in. For higher powers I've no idea.
For n=3 the equation is an elliptic curve and can be handled using
current techniques---in fact one can find all solutions with p prime
and n any multiple of 3 this way. In fact the only one is p=2,n=12,x=181.
This is in fact the largest n for which I know a solution. So
are there _any_ solutions with n>12?
A brute force search found p=2,n=4,x=11 as another solution, and this
is the only other solution I know with n>2. A rather fast loop
in pari convinced me that other than those 2 solutions mentioned
above with p=2 and n>2, there were no other solutions with 3<=n<=20
and p<10^7. I've searched much further with n=5,7 and found nothing.
The only method I knew which would work for general n is the
generalisation of the method (due to Nagell in some form
but which I read about in Cassels' book on local fields)
used to prove the result of Nagell that the only
solutions to x^2+7=2^m in integers x,m>=0 have m<=15.
The equation we're interested in clearly has this as a
special case. Having stared at the method of proof, I convinced
myself that in general one would be able to prove that
for any fixed prime p, there are only finitely many n>=0 for which
x^2+7=8p^n has a solution. But this doesn't seem to be strong
enough to give me a bound at all, let alone one which is uniform in p.
One last observation is of course that for n>2 fixed there will
only be finitely many solutions (by Faltings if you like).
And that's where I stand at the minute. Given that Nagell's result
is in the literature, does anyone know of anything else in the
literature that may help?
Kevin Buzzard