A217277 - OEIS

%I #43 Oct 08 2012 21:18:29

%S 5,9,15,21,27,33,51,57,87,93,123,147,213,297,333,483,753,1053,1347,

%T 2307,2643,3237,4287,6003,10293,12477,14403,18147,26247,58803,74253,

%U 739203,1166883,3801333

%N Numbers n such that n^2-9 is divisible by consecutive primes beginning with 2.

%C Searched to 10^8. If this sequence is finite with 34 terms, then there is only one solution to m! + 9 = n^2: m=6, n=27.

%C Consider the Diophantine equations m^2 - 9 = k*(n^2 - 9), which lead to the Pell-like equations m^2 - k*n^2 = 9*(-k + 1) (1) where the only primes dividing k are in P = {2,3,5,7,11,13,17,19} and such that for every k there is at least one solution (m,n) that belongs to the sequence.

%C As an example, if k = 10 = 2*5, the 2 pairs of solutions of m^2 - 10n^2 = -81; (27,9) and (333,1053) belong to the sequence but the other solutions seem to not be P-smooth. If k = 30 = 2*3*5;  m^2 - 30n^2 = -261 and (147, 27) is a solution  belonging to the sequence, ...

%C If the infinitely many solutions of the Pell-like equations are never P-smooth, then this sequence is finite and there is a precise answer to the extended Brocard's problem: There are exactly 3 solutions to m!+t^2 = n^2 with t=1; 1 solution with t = 3; ... - _Robin Garcia_, Oct 01 2012

%C All terms beyond the first are 3 mod 6, since otherwise n^2 - 9 = 2^k for some k and so n - 3 and n + 3 are both powers of 2. For n > 7 all terms are +-3 mod 30, a consequence of Størmer's theorem. - _Charles R Greathouse IV_, Oct 01 2012

%C No further terms up to 10^10. - _Charles R Greathouse IV_, Oct 01 2012

%e 5^2 - 9 = 2^4; 9^2 -9 = 2^3*3^2; 27^2 -9 = 2^4*3^2*5 = 6!; 87^2 - 9 = 2^3*3^3*5*7; 333^2 - 9 = 2^5*3^2*5*7*11.

%o (PARI) is(n)=my(m=n^2-9,t);forprime(p=2,,t=valuation(m,p);if(t,m/=p^t,return(m==1)))

%o print1(5);forstep(n=9,1e6,6,if(is(n),print1(", "n)))

%o \\ _Charles R Greathouse IV_, Oct 01 2012

%o (PARI) is(n)=my(f=factor(n^2-9)[,1]~);f==primes(#f) \\ _Charles R Greathouse IV_, Oct 01 2012

%Y Cf. A055932, A194099.

%K nonn

%O 1,1

%A _Robin Garcia_, Sep 29 2012