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

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, 21463, 22737, 3, 10, 25, 55, 3, 14, 49, 91, 4, 14, 49, 105, 4, 18, 81, 153, 5, 13, 40, 105, 5, 18, 81, 171, 5, 22, 121, 231, 5, 574, 27881, 30751, 6, 22, 121, 253, 6, 26, 169, 325

Offset: 1

Author

Adam Kertesz and Christopher Hunt Gribble, Sep 25 2015

Keywords

• nonn
• tabf

Comments

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

Of the 224 solutions:

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

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

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

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

(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));

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

Examples

			The sequence is the result of concatenating the rows in the table of solutions for a, x, b and y. The table starts:
.  a      x      b      y
.  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  21463  22737
.  3     10     25     55
.  3     14     49     91
.  4     14     49    105
.  4     18     81    153
.  5     13     40    105
.  5     18     81    171
.  5     22    121    231
.  5    574  27881  30751
.  6     22    121    253
.  6     26    169    325
For a=7 the three solution quadruplets are (7,26,169,351), (7,30,225,435) and (7,1340,120400,129780).
		

Links

• Christopher Hunt Gribble, Table of n, a(n) for n = 1..896

Programs

Maple

for a to 100 do
   for x to 2000000 do
      y := ceil(sqrt(a*x^3));
      b := y^2-a*x^3;
      if b > 0 and y = a*x+b then
         printf("%10d   %10d   %10d   %10d\n", a, x, b, y);
      end if;
   end do;
end do;