<b>The Logical Entropy Change of Euler Infinite Descent</b>

The concept of logical entropy is explained, and the logical entropy principle, which is generally applicable to locating logical loopholes in mathematics, natural sciences, and social sciences, is introduced. The Euler infinite descent method of the cubic Fermat's last conjecture is optimized, and the logical loopholes hidden in Euler's theory due to the change of logical entropy are revealed based on the logical entropy principle. Then two strict statements of the Euler cubic infinite descent method of the cubic Fermat's last conjecture are given to clarify its accurate scope of application. The incomplete quaternion of the spherical Diophantine equation is further constructed, so as to specifically illustrate that the logical entropy change of the solution of the Diophantine equation is accompanied by logical loopholes. It is concluded that the logical entropy change of the Euler infinite descent method leads to logical loopholes, and the possible causal relationship is described as the necessary causal relationship, which actually fails to complete the proof of the cubic Fermat's last conjecture.