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