A model for describing and discovering a mathematical solution to a difficult conjecture:  the case of Fermat’s last theorem. This implies observation, data analysis, heuristics, intuition,  and sometimes serendipity. We shall provide some ideas into finding a solution to the last  theorem. The important thing is not only to follow known methods of proof but also to see the  problem in a different light, and look in many directions. We shall endeavour to suggest a  simple proof for the case of n = 3. Femat’s Last Theorem (FLT) stipulates correctly that there  is no integer solution for zⁿ = xⁿ + yⁿ when n > 2, in terms of z, x, and y whole numbers. The  main search has been to find a counterexample or prove that there is none, so if zⁿ − yⁿ = xⁿ,  and zⁿ − yⁿ / A and B, when A = aⁿ and B = bⁿ, we can make A = 2ⁿ , divide the equation and  find B, and show that when a is an integer, b cannot be an integer, but rather an irrational  number, however for x to be an integer, a and b must b integers, and that is not possible. Moreover we have also found a way of showing that x is irrational.