cp's OEIS Frontend

A232802 Number of solution pairs (x,y) for x <= 11 such that x! + n = y^2 (Brocard-Ramanujan Diophantine equation) is soluble over the integers.

%I A232802 #20 Nov 28 2020 06:19:18
%S A232802 3,1,2,0,0,0,1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0,1,2,1,0,0,0,0,1,0,0,0,1,
%T A232802 1,0,0,0,0,1,0,0,1,0,0,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0,1,1,1,0,0,0,0,
%U A232802 0,0,0,0,0,0,1,2,0,0,1,1,1,0
%N A232802 Number of solution pairs (x,y) for x <= 11 such that x! + n = y^2 (Brocard-Ramanujan Diophantine equation) is soluble over the integers.
%C A232802 The Mathematica program will find the number of integer pairs (x,y) solving x!+n = y^2 for each n from 1 to 200 with x not exceeding 11. Dabrowski showed that the abc conjecture implies only finite solutions for each n. Berndt and Galway found that 11 was the highest value that x reached for a solution with n in the range 1 to 2500 and could find no further solution pairs (x,y) in that range even when x was increased to 10^5.
%C A232802 For n = 1 the number of solutions and arbitrary x is Brocard's problem, and it is conjectured - but verified only in the range x <= 10^12 - that there are 3 solution pairs (x,y): (4,5), (5,11), (7,71). - _Georg Fischer_, Nov 27 2020
%H A232802 Bruce Berndt and William Galway, <a href="http://www.math.uiuc.edu/~berndt/articles/galway.pdf">On the Diophantine equation n! + 1 = m^2</a>
%H A232802 Andrzej Dabrowski, <a href="https://www.researchgate.net/publication/322630341_On_the_Diophantine_Equation_x_Ay_2">On the Diophantine equation x! + A = y^2</a>, Nieuw Archief voor Wiskunde, Vierde serie Deel 14 No. 3 (Nov. 1996) pp. 321-324.
%H A232802 Wikipedia, <a href="https://en.wikipedia.org/wiki/Brocard%27s_problem">Brocard's problem</a>
%H A232802 Wikipedia, <a href="http://en.wikipedia.org/wiki/Abc_conjecture#Some_consequences">abc conjecture - some consequences</a>
%t A232802 Table[Length@Select[Sqrt[Range[11]!+n], IntegerQ[#] &], {n, 1, 200}]
%Y A232802 Cf. A085692, A146968, A216071 (Brocard's problem; all essentially the same sequence).
%Y A232802 Cf. A038202, A068869.
%K A232802 nonn
%O A232802 1,1
%A A232802 _Frank M Jackson_, Nov 30 2013
%E A232802 Definition narrowed by _Georg Fischer_, Nov 27 2020