Consider three positive integers x < y < z in arithmetic sequence, and determine analytically all possible solutions of each of the following equations:

(I) x^{2} + y^{2} = z^{2} - 135

(II) x^{2} + 3y^{2} = z^{2} - 105

(III) x^{2} +y^{2} = z^{2} - xyz

(In reply to

Solution by TamTam)

Your method seems to be accurate, although I solved (I) and (II) by a different method.

However, I do not fully agree with your contention that x = d +sqrt(4*d^2 - 135) for d >= 5.81 gives all possible solutions to (I) while x = 1/3 * ( -d + sqrt(4*d^2 - 315) ) ; for d>=10.247 gives all possible solutions to (II).

For example d= 7, in (I) gives x = 7 + sqrt(61) an irrational quantity which is a contradiction (since x is a positive integer)

Similarly, d = 12, in (II) gives x = 1/3 * ( -12 + sqrt(261)) would lead to a similar contravention of the tenets governing the problem.

For the methodology employed by myself in solving (I) and (II) you can refer to Further Arithmetic Integers (http://perplexus.info/show.php?pid=4729&op=sol), which uses a similar method.

Following this methodology there exits precisely 5 solutions to (I) and 4 solutions to (II)

*Edited on ***January 16, 2007, 11:17 pm**

*Edited on ***January 16, 2007, 11:18 pm**