A328453 Numbers k such that y!/x! + 1 = k^2 has a nonnegative integer solution with y - x <> 1 or 4.

5, 11, 71, 419, 4159

Offset

1,1

Comments

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

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

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

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

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).

Brocard's problem is the case x=1.

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

Links

Table of n, a(n) for n=1..5.

Wikipedia, Brocard's problem

MathOverflow, On a variant of Brocard's problem using the definition of Pochhammer symbols

Example

   4!/1!  + 1 =    5^2.
   6!/3!  + 1 =   11^2.
   7!/1!  + 1 =   71^2.
  57!/54! + 1 =  419^2.
  14!/7!  + 1 = 4159^2.

Maple

N:= 10^15: # to get all terms <= N
L:= N^2-1:
Res:= {5, 11, 419}:
for x from 1 do
  t:= (x+1)*(x+2)*(x+3)*(x+4)*(x+5);
  if t > L then break fi;
  if issqr(t+1) then
    Res:= Res union  {sqrt(t+1)};
  fi;
  for y from x+6 do
      t:= t*y;
      if t > L then break fi;
      if issqr(t+1) then
        Res:= Res union {sqrt(t+1)};
      fi;
  od;
od:
sort(convert(Res, list));

Crossrefs

Cf. A028387.

Sequence in context: A050568 A065555 A189802 * A175207 A067890 A303549

Adjacent sequences: A328450 A328451 A328452 * A328454 A328455 A328456

Keyword

nonn,more

Author

Robert Israel, Oct 15 2019

Status

approved