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A262598 Small positive integer solutions of the simultaneous equations y = ax + b and y^2 = ax^3 + b.

%I A262598 #21 Oct 28 2021 06:27:40
%S A262598 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 A262598 21463,22737,3,10,25,55,3,14,49,91,4,14,49,105,4,18,81,153,5,13,40,
%U A262598 105,5,18,81,171,5,22,121,231,5,574,27881,30751,6,22,121,253,6,26,169,325
%N A262598 Small positive integer solutions of the simultaneous equations y = ax + b and y^2 = ax^3 + b.
%C A262598 The b-file contains all solutions for 1<=a<=100 and 1<=x<=2000000, double-checked by APL and Maple programs.
%C A262598 Of the 224 solutions:
%C A262598 100 are of the form (a, 2(2a-1), (2a-1)^2, (4a-1)(2a-1));
%C A262598 100 are of the form (a, 2(2a+1), (2a+1)^2, (4a+1)(2a+1));
%C A262598 6 are of the form (F(k), F(k+2), F(k)*F(k+1), F(k)*F(k+3)),
%C A262598 where F(n) is the n-th Fibonacci number and k is odd;
%C A262598 (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 A262598 19 have the form (a, cs, c^2t, cu) but appear to be otherwise unrelated.
%H A262598 Christopher Hunt Gribble, <a href="/A262598/b262598.txt">Table of n, a(n) for n = 1..896</a>
%e A262598 The sequence is the result of concatenating the rows in the table of solutions for a, x, b and y. The table starts:
%e A262598 .  a      x      b      y
%e A262598 .  1      2      1      3
%e A262598 .  1      6      9     15
%e A262598 .  2      5      6     16
%e A262598 .  2      6      9     21
%e A262598 .  2     10     25     45
%e A262598 .  2     57    495    609
%e A262598 .  2    637  21463  22737
%e A262598 .  3     10     25     55
%e A262598 .  3     14     49     91
%e A262598 .  4     14     49    105
%e A262598 .  4     18     81    153
%e A262598 .  5     13     40    105
%e A262598 .  5     18     81    171
%e A262598 .  5     22    121    231
%e A262598 .  5    574  27881  30751
%e A262598 .  6     22    121    253
%e A262598 .  6     26    169    325
%e A262598 For a=7 the three solution quadruplets are (7,26,169,351), (7,30,225,435) and (7,1340,120400,129780).
%p A262598 for a to 100 do
%p A262598    for x to 2000000 do
%p A262598       y := ceil(sqrt(a*x^3));
%p A262598       b := y^2-a*x^3;
%p A262598       if b > 0 and y = a*x+b then
%p A262598          printf("%10d   %10d   %10d   %10d\n", a, x, b, y);
%p A262598       end if;
%p A262598    end do;
%p A262598 end do;
%K A262598 nonn,tabf
%O A262598 1,2
%A A262598 _Adam Kertesz_ and _Christopher Hunt Gribble_, Sep 25 2015