From: k.buzzard@ic.ac.uk (Kevin Buzzard) Subject: A fairly-innocent looking equation Date: 2 Feb 99 18:29:40 GMT Newsgroups: sci.math.numberthy Keywords: x^2 + 7 nearly a power? 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 ============================================================================== From: k.buzzard@ic.ac.uk (Kevin Buzzard) Subject: Re: A fairly-innocent looking equation Date: 5 Feb 99 21:02:54 GMT Newsgroups: sci.math.numberthy Hello to all. I recieved several replies about the equation x^2+7=8p^n and all were very gratefully recieved. Apologies to those of you to whom I didn't reply individually. Because I'm not 100% sure who replied to the mailing list and who just mailed me, I thought I'd summarise some of the results I learnt over the last day or two about this equation. Just as a passing remark: it's incredible how the internet can enable someone (e.g. me!) to go from seeing an equation like this to basically knowing the state of the art regarding its solution, in just a few days. A reminder: the "real" question was: give a bound N such that there are no solutions to the equation above, in integers, with p prime and n>N. In my initial posting I made the remark that I wasn't 100% sure whether such a bound should exist, but, as was pointed out to me by several people, the existence of a bound is a consequence of the A-B-C conjecture. So at least one can conjecture that N exists, to begin with. The replies I got were from both algebraists and transcendence theorists (and I had never met most of the transcendence theorists so I'm extra-grateful that they replied!). I guess nowadays there's a very healthy rivalry between these two groups when it comes to Diophantine equations :) I'll summarise what some of the algebraists said first, then some of the transcendence guys afterwards. Jordan Ellenberg clearly had a lot of fun with the equation: he pointed out that if A^2+7=8p^n is a solution, then one can consider the elliptic curve E:y^2=x^3+(3^3*7)x-(2*3^3*7)A. (Other people remarked that the theory of elliptic curves and modular forms might get somewhere, as it did for FLT etc). The point is that this curve has good reduction away from 2,3,7 and p. Furthermore, the reduction is semi-stable at p. Now one can start considering the representation on the q-torsion of E for various primes q and try to get somewhere using the theory of modular forms, for example one could try and prove that only a finite set of primes q could divide n. Unfortunately Jordan couldn't prove existence of N unconditionally. Frits Beukers reminded me of his 1980 paper where he proves that a large class of sequences obtained by recurrence relations of order 2 can only take the value +-1 at most 3 times [See Cassels' exposition of Nagell's proof for p=2 to see why recurrence relations have anything to do with this question]. In fact, for the case in hand, Frits thinks that his method seems to show that, for a _fixed_ p>2, there is at most one possible value of n>0 for which x^2+7=8p^n is solvable with x an integer. I thought this result was fascinating but unfortunately it doesn't answer the question either. Some people gave me some useful buzz-words: "Exponential Diophantine Equation" and "generalized Ramanujan-Nagell equation" were among them. Finally, what can one say about the question using transcendence techniques? Well, actually, a whole lot more! Several people pointed out to me that, in fact, the existence of N _is_ known, effectively and unconditionally, from a theorem of Schinzel and Tijdeman. A reference for all this is Theorem 10.2 of "Exponential Diophantine Equations" by Shorey and Tijdeman. Interestingly enough, the elliptic curve algebric approach might well use the fact that p is prime, but the transcendence approach shows Theorem: Say f is a non-zero poly in Z[X] with at least 2 distinct roots, b is a non-zero integer, m>=0 and y>1 are integers, x is an integer and f(x)=by^m. Then m is less than some effective constant depending only on f,b. This shows the existence of N without even assuming p is prime. One wonders whether one could do any better using transcendence methods and this assumption, but this is certainly something I know very little about. Anyway, the big question left is "What is N"? According to a couple of people, nowadays N is probably about 10^15 although it might be reducible to 1000 or so, using various clever techniques. Anyway, this turns out to be no use for my boss, who has to do calculations in GL_N(Z/pZ) where N is the bound. But basically I guess he _really_ wanted to know whether one could attempt to prove that N=12 was the correct bound nowadays; unfortunately I think that we still can't. Still, at least I had a whole lot of fun! (And learnt a lot too). Thanks a lot to everyone, Kevin Buzzard