Consider three positive integers x < y < z in arithmetic sequence, and determine analytically all possible solutions of each of the following equations:
(I) x2 + y2 = z2 - 135
(II) x2 + 3y2 = z2 - 105
(III) x2 +y2 = z2 - xyz
(In reply to Solution
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