January 23, 2021 Theorem ( ∃ m ∈ N )( ∃ n ∈ N )(3 m + 5 n = 12) Proof : Put the expression in the form 3 m = 12 − 5 n As ( m, n ∈ N ) m and n are both positive, which means 3m is positive also, meaning that (5 n < 12), limiting n to be either 1 or 2. When n = 1, m = 7 3 and 7 3 is not a natural number. When n = 2 , ( m = 2 3 and 2 3 is not a natural number. This proves that due to the positive constraints upon m and n, there is no such combination which satisfies the Theorem. 1