A theorem is a statement to which is subsequently attached a description of the proof of a certain statement or hypothesis known to everyone. The latter can be formulated earlier by some specific author, which should also include the author, whose name is unknown.
The theorem is formulated in order to make it clear which problem is solved in it and in which well-known hypothesis it was formulated earlier or presented for the first time.
The formulation and description of the proof of the theorem are the basis for new hypotheses in which these proofs are used and which, in our case, cannot be related to Fermat’s conjecture.
With regard to Fermat’s theorem, it can be recalled that society knows its formulation, which asserts the existence of evidence for the hypothesis presented in this formulation. But, for certain reasons, society did not receive a description of such evidence.
The presence of only a formulation in which only a description of the essence of the hypothesis is presented cannot be the basis for new hypotheses and, even more so, for references to such non-existent hypotheses that could be the basis for new theorems.
The theorem is the solution. Such a theorem does not need to be solved a second time, since the second solution usually generates a second author of the same product. If the description of the solution is not presented or it is incorrect, then this is a hypothesis and its other solution will be another theorem, regardless of whether the solution presented in it is recognized as true or incorrect.
Recall that at the end of the last century Wiles published his description of the proof of Fermat’s conjecture concerning the solution of equations of the form xn+yn=zn. Certainly this description should be called the theorem of his name, i.e. Wiles’ theorem, since the said description cannot bear any relation to the unknown proof of the said conjecture that Fermat had, according to him.
Let us note that Fermat did not publish his own description of the proof of his conjecture, but argued that he did not present such a description to the public in accordance with certain circumstances and claimed that he had proved the indicated conjecture.
The world community, out of respect for the author of the hypothesis, having no description of the proof of unknown content, recognized its existence and authorship on faith, calling the formulation of this hypothesis Fermat’s theorem.
Later, at the everyday level, the unproven Fermat’s hypothesis in certain interests began to be called a theorem, despite the fact that the content of its proof, as well as the reality of its existence, are unknown. The hypothesis was given the authoritative name “Fermat’s theorem”, embellishing a specialist who has a certain idea about it.
Subsequently, more than one normal proof of this hypothesis may appear, based on completely different achievements of world science. Such concrete proofs will be exclusively the theorems of their authors. Like the unknown proof of the true author of Fermat’s conjecture, or, in other words, the assertion that he has a proof of his hypothesis and, according to this assertion, the unknown proof is recognized as a theorem in the name of this famous and world-respected scientist.
The authors of other correct proofs certainly cannot have anything to do with Fermat’s proof, that is, with his theorem, until the world community itself recognizes the identity of Fermat’s statement of one definite proof from among the new arrivals.
Based on my own experience, I can assume that, figuratively speaking about the fields of the book, Fermat had in mind the simplicity of formulating the theorem according to its mental image, which only a genius can make, with the incredible difficulty of describing its proof for the needs of an ordinary reader.
He certainly felt this complexity, but he could not even imagine its limits, since the life of one person, even the most talented, would not be enough to do this work in the absence of the opportunities provided by modern communications and new computing technology.