A262598 - OEIS

A262598

Small positive integer solutions of the simultaneous equations y = ax + b and y^2 = ax^3 + b.

%I #21 Oct 28 2021 06:27:40

%S 1,2,1,3,1,6,9,15,2,5,6,16,2,6,9,21,2,10,25,45,2,57,495,609,2,637,

%T 21463,22737,3,10,25,55,3,14,49,91,4,14,49,105,4,18,81,153,5,13,40,

%U 105,5,18,81,171,5,22,121,231,5,574,27881,30751,6,22,121,253,6,26,169,325

%N Small positive integer solutions of the simultaneous equations y = ax + b and y^2 = ax^3 + b.

%C The b-file contains all solutions for 1<=a<=100 and 1<=x<=2000000, double-checked by APL and Maple programs.

%C Of the 224 solutions:

%C 100 are of the form (a, 2(2a-1), (2a-1)^2, (4a-1)(2a-1));

%C 100 are of the form (a, 2(2a+1), (2a+1)^2, (4a+1)(2a+1));

%C 6 are of the form (F(k), F(k+2), F(k)*F(k+1), F(k)*F(k+3)),

%C where F(n) is the n-th Fibonacci number and k is odd;

%C (1, 2, 1, 3) is of both forms (a, 2(2a-1), (2a-1)^2, (4a-1)(2a-1)) and (F(k), F(k+2), F(k)*F(k+1), F(k)*F(k+3));

%C 19 have the form (a, cs, c^2t, cu) but appear to be otherwise unrelated.

%H Christopher Hunt Gribble, <a href="/A262598/b262598.txt">Table of n, a(n) for n = 1..896</a>

%e The sequence is the result of concatenating the rows in the table of solutions for a, x, b and y. The table starts:

%e . a x b y

%e . 1 2 1 3

%e . 1 6 9 15

%e . 2 5 6 16

%e . 2 6 9 21

%e . 2 10 25 45

%e . 2 57 495 609

%e . 2 637 21463 22737

%e . 3 10 25 55

%e . 3 14 49 91

%e . 4 14 49 105

%e . 4 18 81 153

%e . 5 13 40 105

%e . 5 18 81 171

%e . 5 22 121 231

%e . 5 574 27881 30751

%e . 6 22 121 253

%e . 6 26 169 325

%e For a=7 the three solution quadruplets are (7,26,169,351), (7,30,225,435) and (7,1340,120400,129780).

%p for a to 100 do

%p for x to 2000000 do

%p y := ceil(sqrt(a*x^3));

%p b := y^2-a*x^3;

%p if b > 0 and y = a*x+b then

%p printf("%10d %10d %10d %10d\n", a, x, b, y);

%p end if;

%p end do;

%K nonn,tabf

%O 1,2

%A _Adam Kertesz_ and _Christopher Hunt Gribble_, Sep 25 2015