:"Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet."
The English translation of Fermat's Latin statement is:<ref>[http://mathworld.wolfram.com/FermatsLastTheorem.htmlFermat's Last Theorem - Wolfram MathWorld (mathworld.wolfram.com)]</ref>
:"It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."
In a series of lectures in 1993, mathematician [[Andrew Wiles]] announced a proof using techniques in algebraic geometry, relying on the nonconstructive [[Axiom of Choice]].<ref name="Occam">http://www.occampress.com/fermat.pdf Page 5</ref> A flaw was found before publication, and Wiles spent a year on fixing the flaw. Then, in September 1994, he and Richard Taylor announced a new version of the proof. However, criticism does continue on the internet.<ref name="Occam" /> Further criticism came from [[Marilyn vos Savant]], known for her very high [[IQ]] and commentary on [[mathematics]], in her column and book.<ref>Ask Marilyn ® by Marilyn vos Savant, Parade Magazine. November 21, 1993</ref><ref>''The World's Most Famous Math Problem: The Proof of Fermat's Last Theorem and Other Mathematical Mysteries'', Marilyn vos Savant. St. Martin's Griffin, 1993</ref> She questioned the use of [[Non-Euclidean geometry]] and the Axiom of Choice, among other points. She retracted her argument in a 1995 addendum to the book.
The Wiles-Taylor proof also makes use of some [[Grothendieck]] tools in cohomological number theory that use an axiom beyond the standard [[Zermelo-Fraenkel|ZFC]] axioms. It is an open question whether these tools can be formalized into a ZFC proof.<ref>Colin Mclarty [http://www.cwru.edu/artsci/phil/Proving_FLT.pdfIs There a “Simple” Proof ofFermat's Last Theorem? Part (1) Introduction and Several New Approaches by Peter Schorer (cwru.edu)]</ref>
Unlike other mathematical breakthroughs, this claimed proof of 1993 has facilitated little, if any, insights or simplifications since then.
==Transcendental Proof==
A devout Catholic, Fermat held a judicial office within the Church, and served as ''parlementaire'' in Toulouse. He was certainly acquainted with the writings of [[St. Thomas Aquinas]] and the evidence of his penetrating intellect. It is entirely possible that the perceived ''impossibility '' of a proof of the theorem Fermat proposed was ''in itself'' a clear demonstration of the realization of the possibility that what is famously called "Fermat's Last Theorem" was ''in itself'' a transcendental proof of the limitations of human intellect and of the limitations of mathematical and scientific methods in the presence of the wisdom and intelligence of the one transcendent [[God]] comprehending of all things subsumed under the Omniscient and Omnipotent Divine Rule. It is indirectly a remarkable proof that man is not infinitely capable ultimately of finally understanding all things, and is not God. Like the [[Christian mysteries]], Fermat's Last Theorem is above the intellect, but not opposed to it.<ref>See the following:
*[http://www.newadvent.org/cathen/10662a.htm Mystery - Catholic Encyclopedia (newadvent.org)] "In conformity with the usage of the inspired writers of the New Testament theologians give the name ''mystery'' to revealed truths that surpass the powers of natural reason.... In its strict sense a mystery is a supernatural truth, one that of its nature lies above the finite intellect."
*[http://catholictheology.info/summa-theologica/summa-part1.php?q=536 A Tour of the Summa - 86. What the Intellect Knows in Material Things (catholictheology.info)] "The human intellect is a created and finite power. Therefore it cannot perfectly know the infinite."