A328453 - OEIS

%I #14 Oct 16 2019 03:28:15

%S 5,11,71,419,4159

%N Numbers k such that y!/x! + 1 = k^2 has a nonnegative integer solution with y - x <> 1 or 4.

%C y-x = 1 is not included because (x+1)!/x! + 1 = x+2.

%C y-x = 4 is not included because (x+4)!/x! + 1 = (x^2+5*x+5)^2 = A028387(x+1)^2 + 1.

%C y-x = 2 is easily seen not to be possible.

%C y-x = 3 corresponds to the elliptic curve (x + 1)*(x + 2)*(x + 3) + 1 = k^2.

%C According to Sage, the only integral points on that curve are (x,k) = (-3,+-1),(-2,+-1),(-1,+-1),(1,+-5),(3,+-11),(54,+-419).

%C Brocard's problem is the case x=1.

%C If a(6) exists, it is greater than 10^16.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Brocard%27s_problem">Brocard's problem</a>

%H MathOverflow, <a href="https://mathoverflow.net/questions/343631/on-a-variant-of-brocards-problem-using-the-definition-of-pochhammer-symbols/343633">On a variant of Brocard's problem using the definition of Pochhammer symbols</a>

%e    4!/1!  + 1 =    5^2.

%e    6!/3!  + 1 =   11^2.

%e    7!/1!  + 1 =   71^2.

%e   57!/54! + 1 =  419^2.

%e   14!/7!  + 1 = 4159^2.

%p N:= 10^15: # to get all terms <= N

%p L:= N^2-1:

%p Res:= {5,11,419}:

%p for x from 1 do

%p   t:= (x+1)*(x+2)*(x+3)*(x+4)*(x+5);

%p   if t > L then break fi;

%p   if issqr(t+1) then

%p     Res:= Res union  {sqrt(t+1)};

%p   fi;

%p   for y from x+6 do

%p       t:= t*y;

%p       if t > L then break fi;

%p       if issqr(t+1) then

%p         Res:= Res union {sqrt(t+1)};

%p       fi;

%p   od;

%p od:

%p sort(convert(Res,list));

%Y Cf. A028387.

%K nonn,more

%O 1,1

%A _Robert Israel_, Oct 15 2019