A393931 - OEIS

%I #18 Mar 16 2026 22:19:21

%S 3,2,17,10,8,1,74,58,9,88,4,28,25,36,16,400,49,784,169,196,217,1240,

%T 4900,1681,47128,14392,625,1296,844,45889,99148,21592,15376,31684,

%U 21904,289,11449,196249,7609,21316,3721,167104,573049

%N a(n) is the least positive integer k such that the elliptic curve y^2 = x^3 - k*x + 1 has exactly n integer solutions with y >= 0.

%C a(44) > 10^6, a(45) = 16384, a(52) = 120409, a(66)=550624.

%C Conjecture: For all n > 3, a(n) is not prime.

%e a(7) = 74 because 74 is the least k such that the elliptic curve y^2 = x^3 - 74*x + 1 has 7 integral solutions with nonnegative y: {{-8,9}, {-3,14}, {0,1}, {9,8}, {12,29}, {28,141}, {1369,50652}}.

%o (Magma)

%o SetClassGroupBounds("GRH");

%o max_n := 15;

%o max_k := 100;

%o sols := [**];

%o for k in [1..max_k] do

%o     Append(~sols, [k, #IntegralPoints(EllipticCurve([0,0,0,-k,1]))]);

%o end for;

%o for n in [1..max_n] do

%o     for sol in sols do

%o         if sol[2] eq n then

%o             printf "a(%o) = %o\n", n, sol[1];

%o             break;

%o         end if;

%o     end for;

%o end for;

%Y Cf. A392144, A392395, A394164.

%K nonn,more

%O 1,1

%A _Zhining Yang_, Mar 03 2026