Re: exponential diophantine equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg62998] Re: exponential diophantine equations*From*: Peter Pein <petsie at dordos.net>*Date*: Sat, 10 Dec 2005 06:03:13 -0500 (EST)*References*: <dnbndq$5ua$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Andrzej Kozlowski schrieb: > I have experimented a little more with this on the train home (thanks > to my faithful PowerBook) and have a few remarks. > The method I sketched below where I used the first 200 primes p to > test if f[k] is a square mod p is an overkill. In fact, it is much > better to use only a few primes at first (perhaps just one) and then > run the function again over the set of primes that get selected, > using different primes, of course. Here is what I did on the train. > First, I redefined the function g as follows: > > g[k_:1, l_] := Function[x, Evaluate[And @@ Prepend[ > Table[test[x, Prime[i]], {i, k, l}], test25[x]]]] > > > Now g takes two arguments, with the first one being optional and by > default 1. So g[200] will still test the first 200 primes, but g > [200,200] will just test the 200th prime. The test for divisibility of > f[k] by 25 is always included. > > Now this is what I did: > > > ls=Select[Range[2000000],g[5000,5000]];//Timing > > {205.797 Second,Null} > ... A slightly faster approach checks only k in Flatten[Table[{6, 26, 46, 47, 66, 86, 99} + 100*n, {n, 0, kmax/100}]]. Experiments show, that this list takes its strongest reduction with the test g[p1,p2], where p1 and p2 are near EulerGamma*PrimePi[kmax] (most probably by accident(?)): First[Timing[ kmax = 10^7; ls = Select[ Flatten[Table[{6, 26, 46, 47, 66, 86, 99} + 100*n, {n, 0, kmax/100}]], g @@ (Floor[PrimePi[kmax]*EulerGamma] + {-2, 2})]; ]] --> 146.641*Second Length[ls] --> 21876 Timing[Select[ls, g[20]]] --> {6.047*Second, {6, 3706947}} unfortunately no new solution; just too weak tests: g[21, 21][3706947] --> False Peter