2 2 On Solutions of the Diophantine equation, x[1] - 3 x[1] x[2] + x[2] = 1 By Shalosh B. Ekhad The following set of increasing positive integers, [x[1], x[2]], are definitely solutions of the diophatine equation 2 2 x[1] - 3 x[1] x[2] + x[2] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = 3 a[1][j - 1] - a[1][j - 2] Subject to the initial conditions a[1][0] = 0, a[1][1] = 1 Or equivalently in terms of the generating function infinity ----- x \ j ------------ = ) a[1][j] x 2 / x - 3 x + 1 ----- j = 0 Then the following increasing , 2, -tuples [x[1], x[2]] = [a[1][j], a[1][1 + j]] for j from 0 to infinity 2 2 are solutions of the diophantine equation, x[1] - 3 x[1] x[2] + x[2] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- This ends thie article that took, 0.978, to create. --------------------------------------------------- --------------------------------------------------- --------------------------------------------------- 2 2 On Solutions of the Diophantine equation, x[1] - 4 x[1] x[2] + x[2] = 1 By Shalosh B. Ekhad The following set of increasing positive integers, [x[1], x[2]], are definitely solutions of the diophatine equation 2 2 x[1] - 4 x[1] x[2] + x[2] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = 4 a[1][j - 1] - a[1][j - 2] Subject to the initial conditions a[1][0] = 0, a[1][1] = 1 Or equivalently in terms of the generating function infinity ----- x \ j ------------ = ) a[1][j] x 2 / x - 4 x + 1 ----- j = 0 Then the following increasing , 2, -tuples [x[1], x[2]] = [a[1][j], a[1][1 + j]] for j from 0 to infinity 2 2 are solutions of the diophantine equation, x[1] - 4 x[1] x[2] + x[2] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- This ends thie article that took, 0.946, to create. --------------------------------------------------- --------------------------------------------------- 2 2 On Solutions of the Diophantine equation, x[1] - 5 x[1] x[2] + x[2] = 1 By Shalosh B. Ekhad The following set of increasing positive integers, [x[1], x[2]], are definitely solutions of the diophatine equation 2 2 x[1] - 5 x[1] x[2] + x[2] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = 5 a[1][j - 1] - a[1][j - 2] Subject to the initial conditions a[1][0] = 0, a[1][1] = 1 Or equivalently in terms of the generating function infinity ----- x \ j ------------ = ) a[1][j] x 2 / x - 5 x + 1 ----- j = 0 Then the following increasing , 2, -tuples [x[1], x[2]] = [a[1][j], a[1][1 + j]] for j from 0 to infinity 2 2 are solutions of the diophantine equation, x[1] - 5 x[1] x[2] + x[2] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- --------------------------------------------------- --------------------------------------------------- This ends thie article that took, 0.946, to create. 2 2 On Solutions of the Diophantine equation, x[1] - 6 x[1] x[2] + x[2] = 1 By Shalosh B. Ekhad The following set of increasing positive integers, [x[1], x[2]], are definitely solutions of the diophatine equation 2 2 x[1] - 6 x[1] x[2] + x[2] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = 6 a[1][j - 1] - a[1][j - 2] Subject to the initial conditions a[1][0] = 0, a[1][1] = 1 Or equivalently in terms of the generating function infinity ----- x \ j ------------ = ) a[1][j] x 2 / x - 6 x + 1 ----- j = 0 Then the following increasing , 2, -tuples [x[1], x[2]] = [a[1][j], a[1][1 + j]] for j from 0 to infinity 2 2 are solutions of the diophantine equation, x[1] - 6 x[1] x[2] + x[2] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- --------------------------------------------------- --------------------------------------------------- This ends thie article that took, 0.959, to create. 2 2 On Solutions of the Diophantine equation, x[1] - 7 x[1] x[2] + x[2] = 1 By Shalosh B. Ekhad The following set of increasing positive integers, [x[1], x[2]], are definitely solutions of the diophatine equation 2 2 x[1] - 7 x[1] x[2] + x[2] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = 7 a[1][j - 1] - a[1][j - 2] Subject to the initial conditions a[1][0] = 0, a[1][1] = 1 Or equivalently in terms of the generating function infinity ----- x \ j ------------ = ) a[1][j] x 2 / x - 7 x + 1 ----- j = 0 Then the following increasing , 2, -tuples [x[1], x[2]] = [a[1][j], a[1][1 + j]] for j from 0 to infinity 2 2 are solutions of the diophantine equation, x[1] - 7 x[1] x[2] + x[2] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- --------------------------------------------------- --------------------------------------------------- This ends thie article that took, 0.948, to create. 2 2 On Solutions of the Diophantine equation, x[1] - 8 x[1] x[2] + x[2] = 1 By Shalosh B. Ekhad The following set of increasing positive integers, [x[1], x[2]], are definitely solutions of the diophatine equation 2 2 x[1] - 8 x[1] x[2] + x[2] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = 8 a[1][j - 1] - a[1][j - 2] Subject to the initial conditions a[1][0] = 0, a[1][1] = 1 Or equivalently in terms of the generating function infinity ----- x \ j ------------ = ) a[1][j] x 2 / x - 8 x + 1 ----- j = 0 Then the following increasing , 2, -tuples [x[1], x[2]] = [a[1][j], a[1][1 + j]] for j from 0 to infinity 2 2 are solutions of the diophantine equation, x[1] - 8 x[1] x[2] + x[2] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- This ends thie article that took, 0.936, to create. --------------------------------------------------- --------------------------------------------------- 2 2 On Solutions of the Diophantine equation, x[1] - 9 x[1] x[2] + x[2] = 1 By Shalosh B. Ekhad The following set of increasing positive integers, [x[1], x[2]], are definitely solutions of the diophatine equation 2 2 x[1] - 9 x[1] x[2] + x[2] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = 9 a[1][j - 1] - a[1][j - 2] Subject to the initial conditions a[1][0] = 0, a[1][1] = 1 Or equivalently in terms of the generating function infinity ----- x \ j ------------ = ) a[1][j] x 2 / x - 9 x + 1 ----- j = 0 Then the following increasing , 2, -tuples [x[1], x[2]] = [a[1][j], a[1][1 + j]] for j from 0 to infinity 2 2 are solutions of the diophantine equation, x[1] - 9 x[1] x[2] + x[2] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- This ends thie article that took, 1.202, to create. 2 2 On Solutions of the Diophantine equation, x[1] - 10 x[1] x[2] + x[2] = 1 By Shalosh B. Ekhad The following set of increasing positive integers, [x[1], x[2]], are definitely solutions of the diophatine equation 2 2 x[1] - 10 x[1] x[2] + x[2] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = 10 a[1][j - 1] - a[1][j - 2] Subject to the initial conditions a[1][0] = 0, a[1][1] = 1 Or equivalently in terms of the generating function infinity ----- x \ j ------------- = ) a[1][j] x 2 / x - 10 x + 1 ----- j = 0 Then the following increasing , 2, -tuples [x[1], x[2]] = [a[1][j], a[1][1 + j]] for j from 0 to infinity 2 2 are solutions of the diophantine equation, x[1] - 10 x[1] x[2] + x[2] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- This ends thie article that took, 1.219, to create. --------------------------------------------------- --------------------------------------------------- ------------------------------------------------- 3 2 2 On Solutions of the Diophantine equation, x[1] + 2 x[1] x[2] + x[1] x[3] 2 2 3 2 + 2 x[1] x[2] - 2 x[1] x[2] x[3] - x[1] x[3] + 2 x[2] - 2 x[2] x[3] 3 + x[3] = 1 By Shalosh B. Ekhad The following set of increasing positive integers, [x[1], x[2], x[3]], are definitely solutions of the diophatine equation 3 2 2 2 x[1] + 2 x[1] x[2] + x[1] x[3] + 2 x[1] x[2] - 2 x[1] x[2] x[3] 2 3 2 3 - x[1] x[3] + 2 x[2] - 2 x[2] x[3] + x[3] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = a[1][j - 1] + a[1][j - 2] + a[1][j - 3] Subject to the initial conditions a[1][0] = 0, a[1][1] = 0, a[1][2] = 1 Or equivalently in terms of the generating function infinity 2 ----- x \ j ---------------- = ) a[1][j] x 3 2 / -x - x - x + 1 ----- j = 0 Then the following increasing , 3, -tuples [x[1], x[2], x[3]] = [a[1][j], a[1][1 + j], a[1][2 + j]] for j from 0 to infinity 3 2 2 are solutions of the diophantine equation, x[1] + 2 x[1] x[2] + x[1] x[3] 2 2 3 2 + 2 x[1] x[2] - 2 x[1] x[2] x[3] - x[1] x[3] + 2 x[2] - 2 x[2] x[3] 3 + x[3] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- This ends thie article that took, 395.761, to create. --------------------------------------------------- --------------------------------------------------- 3 2 2 On Solutions of the Diophantine equation, x[1] + 4 x[1] x[2] + x[1] x[3] 2 2 3 2 + 5 x[1] x[2] - x[1] x[2] x[3] - 2 x[1] x[3] + 3 x[2] - x[2] x[3] 2 3 - 2 x[2] x[3] + x[3] = 1 By Shalosh B. Ekhad The following, 2, sets of increasing positive integers, [x[1], x[2], x[3]], are definitely solutions of the diophatine equation 3 2 2 2 x[1] + 4 x[1] x[2] + x[1] x[3] + 5 x[1] x[2] - x[1] x[2] x[3] 2 3 2 2 3 - 2 x[1] x[3] + 3 x[2] - x[2] x[3] - 2 x[2] x[3] + x[3] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = a[1][j - 1] + 2 a[1][j - 2] + a[1][j - 3] Subject to the initial conditions a[1][0] = 0, a[1][1] = 0, a[1][2] = 1 Or equivalently in terms of the generating function infinity 2 ----- x \ j ------------------ = ) a[1][j] x 3 2 / -x - 2 x - x + 1 ----- j = 0 Then the following increasing , 3, -tuples [x[1], x[2], x[3]] = [a[1][j], a[1][1 + j], a[1][2 + j]] for j from 0 to infinity 3 2 2 are solutions of the diophantine equation, x[1] + 4 x[1] x[2] + x[1] x[3] 2 2 3 2 + 5 x[1] x[2] - x[1] x[2] x[3] - 2 x[1] x[3] + 3 x[2] - x[2] x[3] 2 3 - 2 x[2] x[3] + x[3] = 1 ----------------------------------------------------- Define the sequence , a[2][j], for j from 1 to infinity as the solution of the recurrence a[2][j] = a[2][j - 1] + 2 a[2][j - 2] + a[2][j - 3] Subject to the initial conditions a[2][0] = 0, a[2][1] = 1, a[2][2] = 2 Or equivalently in terms of the generating function infinity 2 ----- x + x \ j ------------------ = ) a[2][j] x 3 2 / -x - 2 x - x + 1 ----- j = 0 Then the following increasing , 3, -tuples [x[1], x[2], x[3]] = [a[2][j], a[2][1 + j], a[2][2 + j]] for j from 0 to infinity 3 2 2 are solutions of the diophantine equation, x[1] + 4 x[1] x[2] + x[1] x[3] 2 2 3 2 + 5 x[1] x[2] - x[1] x[2] x[3] - 2 x[1] x[3] + 3 x[2] - x[2] x[3] 2 3 - 2 x[2] x[3] + x[3] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- This ends thie article that took, 386.859, to create. --------------------------------------------------- --------------------------------------------------- 3 2 2 On Solutions of the Diophantine equation, x[1] + 6 x[1] x[2] + x[1] x[3] 2 2 3 2 2 + 10 x[1] x[2] - 3 x[1] x[3] + 4 x[2] - 2 x[2] x[3] - 2 x[2] x[3] 3 + x[3] = 1 By Shalosh B. Ekhad The following set of increasing positive integers, [x[1], x[2], x[3]], are definitely solutions of the diophatine equation 3 2 2 2 2 3 x[1] + 6 x[1] x[2] + x[1] x[3] + 10 x[1] x[2] - 3 x[1] x[3] + 4 x[2] 2 2 3 - 2 x[2] x[3] - 2 x[2] x[3] + x[3] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = a[1][j - 1] + 3 a[1][j - 2] + a[1][j - 3] Subject to the initial conditions a[1][0] = 0, a[1][1] = 0, a[1][2] = 1 Or equivalently in terms of the generating function infinity 2 ----- x \ j ------------------ = ) a[1][j] x 3 2 / -x - 3 x - x + 1 ----- j = 0 Then the following increasing , 3, -tuples [x[1], x[2], x[3]] = [a[1][j], a[1][1 + j], a[1][2 + j]] for j from 0 to infinity 3 2 2 are solutions of the diophantine equation, x[1] + 6 x[1] x[2] + x[1] x[3] 2 2 3 2 2 + 10 x[1] x[2] - 3 x[1] x[3] + 4 x[2] - 2 x[2] x[3] - 2 x[2] x[3] 3 + x[3] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- This ends thie article that took, 369.960, to create. --------------------------------------------------- --------------------------------------------------- 3 2 2 On Solutions of the Diophantine equation, x[1] + 2 x[1] x[2] + 2 x[1] x[3] 2 2 3 2 + 3 x[1] x[2] - x[1] x[2] x[3] - x[1] x[3] + 3 x[2] + 3 x[2] x[3] 2 3 - 4 x[2] x[3] + x[3] = 1 By Shalosh B. Ekhad The following set of increasing positive integers, [x[1], x[2], x[3]], are definitely solutions of the diophatine equation 3 2 2 2 x[1] + 2 x[1] x[2] + 2 x[1] x[3] + 3 x[1] x[2] - x[1] x[2] x[3] 2 3 2 2 3 - x[1] x[3] + 3 x[2] + 3 x[2] x[3] - 4 x[2] x[3] + x[3] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = 2 a[1][j - 1] + a[1][j - 2] + a[1][j - 3] Subject to the initial conditions a[1][0] = 0, a[1][1] = 0, a[1][2] = 1 Or equivalently in terms of the generating function infinity 2 ----- x \ j ------------------ = ) a[1][j] x 3 2 / -x - x - 2 x + 1 ----- j = 0 Then the following increasing , 3, -tuples [x[1], x[2], x[3]] = [a[1][j], a[1][1 + j], a[1][2 + j]] for j from 0 to infinity 3 2 2 are solutions of the diophantine equation, x[1] + 2 x[1] x[2] + 2 x[1] x[3] 2 2 3 2 + 3 x[1] x[2] - x[1] x[2] x[3] - x[1] x[3] + 3 x[2] + 3 x[2] x[3] 2 3 - 4 x[2] x[3] + x[3] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- This ends thie article that took, 374.783, to create. --------------------------------------------------- --------------------------------------------------- 3 2 2 On Solutions of the Diophantine equation, x[1] + 4 x[1] x[2] + 2 x[1] x[3] 2 2 3 2 + 6 x[1] x[2] + x[1] x[2] x[3] - 2 x[1] x[3] + 5 x[2] + 2 x[2] x[3] 2 3 - 4 x[2] x[3] + x[3] = 1 By Shalosh B. Ekhad The following set of increasing positive integers, [x[1], x[2], x[3]], are definitely solutions of the diophatine equation 3 2 2 2 x[1] + 4 x[1] x[2] + 2 x[1] x[3] + 6 x[1] x[2] + x[1] x[2] x[3] 2 3 2 2 3 - 2 x[1] x[3] + 5 x[2] + 2 x[2] x[3] - 4 x[2] x[3] + x[3] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = 2 a[1][j - 1] + 2 a[1][j - 2] + a[1][j - 3] Subject to the initial conditions a[1][0] = 0, a[1][1] = 0, a[1][2] = 1 Or equivalently in terms of the generating function infinity 2 ----- x \ j -------------------- = ) a[1][j] x 3 2 / -x - 2 x - 2 x + 1 ----- j = 0 Then the following increasing , 3, -tuples [x[1], x[2], x[3]] = [a[1][j], a[1][1 + j], a[1][2 + j]] for j from 0 to infinity 3 2 2 are solutions of the diophantine equation, x[1] + 4 x[1] x[2] + 2 x[1] x[3] 2 2 3 2 + 6 x[1] x[2] + x[1] x[2] x[3] - 2 x[1] x[3] + 5 x[2] + 2 x[2] x[3] 2 3 - 4 x[2] x[3] + x[3] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- This ends thie article that took, 384.821, to create. --------------------------------------------------- --------------------------------------------------- 3 2 2 On Solutions of the Diophantine equation, x[1] + 6 x[1] x[2] + 2 x[1] x[3] 2 2 3 2 + 11 x[1] x[2] + 3 x[1] x[2] x[3] - 3 x[1] x[3] + 7 x[2] + x[2] x[3] 2 3 - 4 x[2] x[3] + x[3] = 1 By Shalosh B. Ekhad The following, 4, sets of increasing positive integers, [x[1], x[2], x[3]], are definitely solutions of the diophatine equation 3 2 2 2 x[1] + 6 x[1] x[2] + 2 x[1] x[3] + 11 x[1] x[2] + 3 x[1] x[2] x[3] 2 3 2 2 3 - 3 x[1] x[3] + 7 x[2] + x[2] x[3] - 4 x[2] x[3] + x[3] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = 2 a[1][j - 1] + 3 a[1][j - 2] + a[1][j - 3] Subject to the initial conditions a[1][0] = 0, a[1][1] = 0, a[1][2] = 1 Or equivalently in terms of the generating function infinity 2 ----- x \ j -------------------- = ) a[1][j] x 3 2 / -x - 3 x - 2 x + 1 ----- j = 0 Then the following increasing , 3, -tuples [x[1], x[2], x[3]] = [a[1][j], a[1][1 + j], a[1][2 + j]] for j from 0 to infinity 3 2 2 are solutions of the diophantine equation, x[1] + 6 x[1] x[2] + 2 x[1] x[3] 2 2 3 2 + 11 x[1] x[2] + 3 x[1] x[2] x[3] - 3 x[1] x[3] + 7 x[2] + x[2] x[3] 2 3 - 4 x[2] x[3] + x[3] = 1 ----------------------------------------------------- Define the sequence , a[2][j], for j from 1 to infinity as the solution of the recurrence a[2][j] = 2 a[2][j - 1] + 3 a[2][j - 2] + a[2][j - 3] Subject to the initial conditions a[2][0] = 0, a[2][1] = 1, a[2][2] = 3 Or equivalently in terms of the generating function infinity 2 ----- x + x \ j -------------------- = ) a[2][j] x 3 2 / -x - 3 x - 2 x + 1 ----- j = 0 Then the following increasing , 3, -tuples [x[1], x[2], x[3]] = [a[2][j], a[2][1 + j], a[2][2 + j]] for j from 0 to infinity 3 2 2 are solutions of the diophantine equation, x[1] + 6 x[1] x[2] + 2 x[1] x[3] 2 2 3 2 + 11 x[1] x[2] + 3 x[1] x[2] x[3] - 3 x[1] x[3] + 7 x[2] + x[2] x[3] 2 3 - 4 x[2] x[3] + x[3] = 1 ----------------------------------------------------- Define the sequence , a[3][j], for j from 1 to infinity as the solution of the recurrence a[3][j] = 2 a[3][j - 1] + 3 a[3][j - 2] + a[3][j - 3] Subject to the initial conditions a[3][0] = 0, a[3][1] = 2, a[3][2] = 5 Or equivalently in terms of the generating function infinity 2 ----- x + 2 x \ j -------------------- = ) a[3][j] x 3 2 / -x - 3 x - 2 x + 1 ----- j = 0 Then the following increasing , 3, -tuples [x[1], x[2], x[3]] = [a[3][j], a[3][1 + j], a[3][2 + j]] for j from 0 to infinity 3 2 2 are solutions of the diophantine equation, x[1] + 6 x[1] x[2] + 2 x[1] x[3] 2 2 3 2 + 11 x[1] x[2] + 3 x[1] x[2] x[3] - 3 x[1] x[3] + 7 x[2] + x[2] x[3] 2 3 - 4 x[2] x[3] + x[3] = 1 ----------------------------------------------------- Define the sequence , a[4][j], for j from 1 to infinity as the solution of the recurrence a[4][j] = 2 a[4][j - 1] + 3 a[4][j - 2] + a[4][j - 3] Subject to the initial conditions a[4][0] = 1, a[4][1] = 1, a[4][2] = 4 Or equivalently in terms of the generating function infinity 2 ----- -x - x + 1 \ j -------------------- = ) a[4][j] x 3 2 / -x - 3 x - 2 x + 1 ----- j = 0 Then the following increasing , 3, -tuples [x[1], x[2], x[3]] = [a[4][j], a[4][1 + j], a[4][2 + j]] for j from 0 to infinity 3 2 2 are solutions of the diophantine equation, x[1] + 6 x[1] x[2] + 2 x[1] x[3] 2 2 3 2 + 11 x[1] x[2] + 3 x[1] x[2] x[3] - 3 x[1] x[3] + 7 x[2] + x[2] x[3] 2 3 - 4 x[2] x[3] + x[3] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- This ends thie article that took, 385.918, to create. --------------------------------------------------- --------------------------------------------------- 3 2 2 On Solutions of the Diophantine equation, x[1] + 8 x[1] x[2] + 2 x[1] x[3] 2 2 3 2 + 18 x[1] x[2] + 5 x[1] x[2] x[3] - 4 x[1] x[3] + 9 x[2] - 4 x[2] x[3] 3 + x[3] = 1 By Shalosh B. Ekhad The following set of increasing positive integers, [x[1], x[2], x[3]], are definitely solutions of the diophatine equation 3 2 2 2 x[1] + 8 x[1] x[2] + 2 x[1] x[3] + 18 x[1] x[2] + 5 x[1] x[2] x[3] 2 3 2 3 - 4 x[1] x[3] + 9 x[2] - 4 x[2] x[3] + x[3] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = 2 a[1][j - 1] + 4 a[1][j - 2] + a[1][j - 3] Subject to the initial conditions a[1][0] = 0, a[1][1] = 0, a[1][2] = 1 Or equivalently in terms of the generating function infinity 2 ----- x \ j -------------------- = ) a[1][j] x 3 2 / -x - 4 x - 2 x + 1 ----- j = 0 Then the following increasing , 3, -tuples [x[1], x[2], x[3]] = [a[1][j], a[1][1 + j], a[1][2 + j]] for j from 0 to infinity 3 2 2 are solutions of the diophantine equation, x[1] + 8 x[1] x[2] + 2 x[1] x[3] 2 2 3 2 + 18 x[1] x[2] + 5 x[1] x[2] x[3] - 4 x[1] x[3] + 9 x[2] - 4 x[2] x[3] 3 + x[3] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- This ends thie article that took, 377.125, to create. --------------------------------------------------- --------------------------------------------------- 3 2 2 On Solutions of the Diophantine equation, x[1] + 2 x[1] x[2] + 3 x[1] x[3] 2 2 3 2 2 + 4 x[1] x[2] - x[1] x[3] + 4 x[2] + 8 x[2] x[3] - 6 x[2] x[3] 3 + x[3] = 1 By Shalosh B. Ekhad The following set of increasing positive integers, [x[1], x[2], x[3]], are definitely solutions of the diophatine equation 3 2 2 2 2 3 x[1] + 2 x[1] x[2] + 3 x[1] x[3] + 4 x[1] x[2] - x[1] x[3] + 4 x[2] 2 2 3 + 8 x[2] x[3] - 6 x[2] x[3] + x[3] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = 3 a[1][j - 1] + a[1][j - 2] + a[1][j - 3] Subject to the initial conditions a[1][0] = 0, a[1][1] = 0, a[1][2] = 1 Or equivalently in terms of the generating function infinity 2 ----- x \ j ------------------ = ) a[1][j] x 3 2 / -x - x - 3 x + 1 ----- j = 0 Then the following increasing , 3, -tuples [x[1], x[2], x[3]] = [a[1][j], a[1][1 + j], a[1][2 + j]] for j from 0 to infinity 3 2 2 are solutions of the diophantine equation, x[1] + 2 x[1] x[2] + 3 x[1] x[3] 2 2 3 2 2 + 4 x[1] x[2] - x[1] x[3] + 4 x[2] + 8 x[2] x[3] - 6 x[2] x[3] 3 + x[3] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- This ends thie article that took, 387.058, to create. --------------------------------------------------- --------------------------------------------------- 3 2 2 On Solutions of the Diophantine equation, x[1] + 4 x[1] x[2] + 3 x[1] x[3] 2 2 3 2 + 7 x[1] x[2] + 3 x[1] x[2] x[3] - 2 x[1] x[3] + 7 x[2] + 7 x[2] x[3] 2 3 - 6 x[2] x[3] + x[3] = 1 By Shalosh B. Ekhad The following set of increasing positive integers, [x[1], x[2], x[3]], are definitely solutions of the diophatine equation 3 2 2 2 x[1] + 4 x[1] x[2] + 3 x[1] x[3] + 7 x[1] x[2] + 3 x[1] x[2] x[3] 2 3 2 2 3 - 2 x[1] x[3] + 7 x[2] + 7 x[2] x[3] - 6 x[2] x[3] + x[3] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = 3 a[1][j - 1] + 2 a[1][j - 2] + a[1][j - 3] Subject to the initial conditions a[1][0] = 0, a[1][1] = 0, a[1][2] = 1 Or equivalently in terms of the generating function infinity 2 ----- x \ j -------------------- = ) a[1][j] x 3 2 / -x - 2 x - 3 x + 1 ----- j = 0 Then the following increasing , 3, -tuples [x[1], x[2], x[3]] = [a[1][j], a[1][1 + j], a[1][2 + j]] for j from 0 to infinity 3 2 2 are solutions of the diophantine equation, x[1] + 4 x[1] x[2] + 3 x[1] x[3] 2 2 3 2 + 7 x[1] x[2] + 3 x[1] x[2] x[3] - 2 x[1] x[3] + 7 x[2] + 7 x[2] x[3] 2 3 - 6 x[2] x[3] + x[3] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- This ends thie article that took, 374.654, to create. --------------------------------------------------- --------------------------------------------------- 3 2 2 On Solutions of the Diophantine equation, x[1] + 6 x[1] x[2] + 3 x[1] x[3] 2 2 3 + 12 x[1] x[2] + 6 x[1] x[2] x[3] - 3 x[1] x[3] + 10 x[2] 2 2 3 + 6 x[2] x[3] - 6 x[2] x[3] + x[3] = 1 By Shalosh B. Ekhad The following set of increasing positive integers, [x[1], x[2], x[3]], are definitely solutions of the diophatine equation 3 2 2 2 x[1] + 6 x[1] x[2] + 3 x[1] x[3] + 12 x[1] x[2] + 6 x[1] x[2] x[3] 2 3 2 2 3 - 3 x[1] x[3] + 10 x[2] + 6 x[2] x[3] - 6 x[2] x[3] + x[3] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = 3 a[1][j - 1] + 3 a[1][j - 2] + a[1][j - 3] Subject to the initial conditions a[1][0] = 0, a[1][1] = 0, a[1][2] = 1 Or equivalently in terms of the generating function infinity 2 ----- x \ j -------------------- = ) a[1][j] x 3 2 / -x - 3 x - 3 x + 1 ----- j = 0 Then the following increasing , 3, -tuples [x[1], x[2], x[3]] = [a[1][j], a[1][1 + j], a[1][2 + j]] for j from 0 to infinity 3 2 2 are solutions of the diophantine equation, x[1] + 6 x[1] x[2] + 3 x[1] x[3] 2 2 3 + 12 x[1] x[2] + 6 x[1] x[2] x[3] - 3 x[1] x[3] + 10 x[2] 2 2 3 + 6 x[2] x[3] - 6 x[2] x[3] + x[3] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- This ends thie article that took, 391.724, to create. --------------------------------------------------- --------------------------------------------------- 3 2 2 On Solutions of the Diophantine equation, x[1] + 2 x[1] x[2] + 4 x[1] x[3] 2 2 3 2 + 5 x[1] x[2] + x[1] x[2] x[3] - x[1] x[3] + 5 x[2] + 15 x[2] x[3] 2 3 - 8 x[2] x[3] + x[3] = 1 By Shalosh B. Ekhad The following set of increasing positive integers, [x[1], x[2], x[3]], are definitely solutions of the diophatine equation 3 2 2 2 x[1] + 2 x[1] x[2] + 4 x[1] x[3] + 5 x[1] x[2] + x[1] x[2] x[3] 2 3 2 2 3 - x[1] x[3] + 5 x[2] + 15 x[2] x[3] - 8 x[2] x[3] + x[3] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = 4 a[1][j - 1] + a[1][j - 2] + a[1][j - 3] Subject to the initial conditions a[1][0] = 0, a[1][1] = 0, a[1][2] = 1 Or equivalently in terms of the generating function infinity 2 ----- x \ j ------------------ = ) a[1][j] x 3 2 / -x - x - 4 x + 1 ----- j = 0 Then the following increasing , 3, -tuples [x[1], x[2], x[3]] = [a[1][j], a[1][1 + j], a[1][2 + j]] for j from 0 to infinity 3 2 2 are solutions of the diophantine equation, x[1] + 2 x[1] x[2] + 4 x[1] x[3] 2 2 3 2 + 5 x[1] x[2] + x[1] x[2] x[3] - x[1] x[3] + 5 x[2] + 15 x[2] x[3] 2 3 - 8 x[2] x[3] + x[3] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- This ends thie article that took, 389.125, to create. --------------------------------------------------- --------------------------------------------------- 3 2 2 On Solutions of the Diophantine equation, x[1] + 4 x[1] x[2] + 4 x[1] x[3] 2 2 3 2 + 8 x[1] x[2] + 5 x[1] x[2] x[3] - 2 x[1] x[3] + 9 x[2] + 14 x[2] x[3] 2 3 - 8 x[2] x[3] + x[3] = 1 By Shalosh B. Ekhad The following set of increasing positive integers, [x[1], x[2], x[3]], are definitely solutions of the diophatine equation 3 2 2 2 x[1] + 4 x[1] x[2] + 4 x[1] x[3] + 8 x[1] x[2] + 5 x[1] x[2] x[3] 2 3 2 2 3 - 2 x[1] x[3] + 9 x[2] + 14 x[2] x[3] - 8 x[2] x[3] + x[3] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = 4 a[1][j - 1] + 2 a[1][j - 2] + a[1][j - 3] Subject to the initial conditions a[1][0] = 0, a[1][1] = 0, a[1][2] = 1 Or equivalently in terms of the generating function infinity 2 ----- x \ j -------------------- = ) a[1][j] x 3 2 / -x - 2 x - 4 x + 1 ----- j = 0 Then the following increasing , 3, -tuples [x[1], x[2], x[3]] = [a[1][j], a[1][1 + j], a[1][2 + j]] for j from 0 to infinity 3 2 2 are solutions of the diophantine equation, x[1] + 4 x[1] x[2] + 4 x[1] x[3] 2 2 3 2 + 8 x[1] x[2] + 5 x[1] x[2] x[3] - 2 x[1] x[3] + 9 x[2] + 14 x[2] x[3] 2 3 - 8 x[2] x[3] + x[3] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- This ends thie article that took, 379.751, to create. --------------------------------------------------- --------------------------------------------------- 3 2 2 On Solutions of the Diophantine equation, x[1] + 6 x[1] x[2] + 4 x[1] x[3] 2 2 3 + 13 x[1] x[2] + 9 x[1] x[2] x[3] - 3 x[1] x[3] + 13 x[2] 2 2 3 + 13 x[2] x[3] - 8 x[2] x[3] + x[3] = 1 By Shalosh B. Ekhad The following set of increasing positive integers, [x[1], x[2], x[3]], are definitely solutions of the diophatine equation 3 2 2 2 x[1] + 6 x[1] x[2] + 4 x[1] x[3] + 13 x[1] x[2] + 9 x[1] x[2] x[3] 2 3 2 2 3 - 3 x[1] x[3] + 13 x[2] + 13 x[2] x[3] - 8 x[2] x[3] + x[3] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = 4 a[1][j - 1] + 3 a[1][j - 2] + a[1][j - 3] Subject to the initial conditions a[1][0] = 0, a[1][1] = 0, a[1][2] = 1 Or equivalently in terms of the generating function infinity 2 ----- x \ j -------------------- = ) a[1][j] x 3 2 / -x - 3 x - 4 x + 1 ----- j = 0 Then the following increasing , 3, -tuples [x[1], x[2], x[3]] = [a[1][j], a[1][1 + j], a[1][2 + j]] for j from 0 to infinity 3 2 2 are solutions of the diophantine equation, x[1] + 6 x[1] x[2] + 4 x[1] x[3] 2 2 3 + 13 x[1] x[2] + 9 x[1] x[2] x[3] - 3 x[1] x[3] + 13 x[2] 2 2 3 + 13 x[2] x[3] - 8 x[2] x[3] + x[3] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- This ends thie article that took, 378.978, to create. --------------------------------------------------- --------------------------------------------------- 3 2 2 On Solutions of the Diophantine equation, x[1] + 8 x[1] x[2] + 4 x[1] x[3] 2 2 3 + 20 x[1] x[2] + 13 x[1] x[2] x[3] - 4 x[1] x[3] + 17 x[2] 2 2 3 + 12 x[2] x[3] - 8 x[2] x[3] + x[3] = 1 By Shalosh B. Ekhad The following, 2, sets of increasing positive integers, [x[1], x[2], x[3]], are definitely solutions of the diophatine equation 3 2 2 2 x[1] + 8 x[1] x[2] + 4 x[1] x[3] + 20 x[1] x[2] + 13 x[1] x[2] x[3] 2 3 2 2 3 - 4 x[1] x[3] + 17 x[2] + 12 x[2] x[3] - 8 x[2] x[3] + x[3] = 1 ----------------------------------------------------- Define the sequence , a[1][j], for j from 1 to infinity as the solution of the recurrence a[1][j] = 4 a[1][j - 1] + 4 a[1][j - 2] + a[1][j - 3] Subject to the initial conditions a[1][0] = 0, a[1][1] = 0, a[1][2] = 1 Or equivalently in terms of the generating function infinity 2 ----- x \ j -------------------- = ) a[1][j] x 3 2 / -x - 4 x - 4 x + 1 ----- j = 0 Then the following increasing , 3, -tuples [x[1], x[2], x[3]] = [a[1][j], a[1][1 + j], a[1][2 + j]] for j from 0 to infinity 3 2 2 are solutions of the diophantine equation, x[1] + 8 x[1] x[2] + 4 x[1] x[3] 2 2 3 + 20 x[1] x[2] + 13 x[1] x[2] x[3] - 4 x[1] x[3] + 17 x[2] 2 2 3 + 12 x[2] x[3] - 8 x[2] x[3] + x[3] = 1 ----------------------------------------------------- Define the sequence , a[2][j], for j from 1 to infinity as the solution of the recurrence a[2][j] = 4 a[2][j - 1] + 4 a[2][j - 2] + a[2][j - 3] Subject to the initial conditions a[2][0] = 0, a[2][1] = 2, a[2][2] = 9 Or equivalently in terms of the generating function infinity 2 ----- x + 2 x \ j -------------------- = ) a[2][j] x 3 2 / -x - 4 x - 4 x + 1 ----- j = 0 Then the following increasing , 3, -tuples [x[1], x[2], x[3]] = [a[2][j], a[2][1 + j], a[2][2 + j]] for j from 0 to infinity 3 2 2 are solutions of the diophantine equation, x[1] + 8 x[1] x[2] + 4 x[1] x[3] 2 2 3 + 20 x[1] x[2] + 13 x[1] x[2] x[3] - 4 x[1] x[3] + 17 x[2] 2 2 3 + 12 x[2] x[3] - 8 x[2] x[3] + x[3] = 1 Furthermore, we conjecture that these are all the increasing positive soluti\ ons ----------------------------------------- This ends thie article that took, 389.276, to create. -------------------------------------------------