Euler Perfect Box (Final)

Abstract of Euler Perfect Box (Final) a, b, c, d, e, f, g ∈ N_+ ⟺ Euler Perfect Box Let a, b, c, d, e, f, r, k ∈N_+. I have to prove that g ∈N_+ Let a < b < c, & [(a < r< k), or (a > r, and r < k)] ⇒ a ≠ r ≠ k & let b=a + r, c = a+ k g ^2=a^2 +b^2 +c^2 ∴ g ^2=a^2 +(a + r) ^2 +(a + k) ^2 Let g^2= [a^2 + (a + r) ^2 +(a + 2] ∈NS) ∵ [a^2+2a (a + k) + (a + k) ^2] ∈NS) ∴ (a + r) ^2= 2a (a + k) ⇔ a = r = k …Contradiction to a≠ r ≠k ⇒ (a + r) ^2 ≠ 2(a + k) for all a, r, k ∈N+)/{a=r=k}. ∴g ^2 = [a^2 +(a + r) ^2 +(a + k) ^2] ∉ N_ (S) ⇒ g^2 ∉ NS) ⇒g ∉ N+) ∴ The Euler Perfect Box does not exist