What is special about the numbers, (3,4,5)?. Answer is, they are the solution to the Pythagoras Equation(PE), x^2 + y^2 = z^2. There are many such integer triplets which are solutions to PE. This is well known since the times of Pythagoras, but, what I am presenting next are new discoveries. Read on.......
Let us generalize PE. Following is a generalization of PE, (a.x^2+b.x+c) + (a.y^2+b.y+c) = (a.z^2+b.z+c), where a,b.c are real constants. Let us take one possible set of values for (a,b,c) = (1,1,0), which gives the equation, (x^2+x) + (y^2+y) = (z^2+z). This equation has the following integer solution, (13,90,91), which can be checked as, (13^2+13) + (90^2+90) = 8372 = (91^2+91). One can easily check that the following triplets are also the solutions of (x^2+x) + (y^2+y) = (z^2+z) : (14,104,105), (15,119,120), (16,135,136), (17,152,153), just to name a few.
Here are some more examples,
(1). (x^2 + x + 2) + (y^2 + y + 2) = (z^2 + z + 2) has the following solutions, (13, 91, 92), (14, 105, 106), (15, 120, 121) just to name a few.
(2). (x^2 + 3.x + 4) + (y^2 + 3.y + 4) = (z^2 + 3.z + 4) has the following solutions, (12,90,91), (13,104,105), (14, 119, 120), (21, 252, 253) just to name a few.
(3). (x^2 + 5.x + 2) + (y^2 + 5.y + 2) = (z^2 + 5.z + 2) has the following solutions, (11,86,87), (12,100,101), (13,115,116), (14,131,132) just to name a few.
(4). (x^2 + 15.x + 4) + (y^2 + 15.y + 4) = (z^2 + 15.z + 4) has the following solutions, (8,14,18), (9,18,22), (16,53,57), (17,59,63) just to name a few.
There are lot many more examples possible. But, I stop here. If it is of any mathematical value to you, contact me at naveenck.mangaloreuniversity@gmail.com, I can provide many more examples with solutions.