Difference between revisions of "Fermat's Last Theorem"
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[[Image:Pierre de Fermat.jpg|thumb|right|Pierre de Fermat]] | [[Image:Pierre de Fermat.jpg|thumb|right|Pierre de Fermat]] | ||
| − | Fermat's Last Theorem asserts that the well-known [[Pythagorean Theorem]] has no solutions for higher powers. That is, | + | '''Fermat's Last Theorem''' asserts that the well-known [[Pythagorean Theorem]] has no solutions for higher powers. That is, |
:<math>a^2 + b^2 = c^2</math> | :<math>a^2 + b^2 = c^2</math> | ||
| − | has | + | has infinitely many solutions, such as {3, 4, 5} or {5, 12, 13}; but |
:<math>a^3 + b^3 = c^3</math> | :<math>a^3 + b^3 = c^3</math> | ||
| − | has no solutions using whole numbers | + | has no solutions using positive whole numbers. |
It was conjectured by the French mathematician [[Pierre de Fermat]]. He said he had proved this problem but that there was not enough room in the margin to state his proof:<ref>Nagell 1951, p. 252.</ref> | It was conjectured by the French mathematician [[Pierre de Fermat]]. He said he had proved this problem but that there was not enough room in the margin to state his proof:<ref>Nagell 1951, p. 252.</ref> | ||
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:"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." | :"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.html</ref> | + | The English translation of Fermat's Latin statement is:<ref>[http://mathworld.wolfram.com/FermatsLastTheorem.html Fermat'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." | :"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." | ||
| − | This problem has confounded mathematicians for centuries, and there still is no proof for it using [[elementary techniques]]. [[Gauss]] and other mathematicians doubt that Fermat was able to prove it himself, but [[Cal Tech]] mathematics Professor E.T. Bell, who wrote the standard biography of all the great mathematicians, wryly observed that "the fox who could not get at the grapes declared they were sour." <ref>E.T. Bell, "Men of Mathematics" 72 (1937).</ref> "And so for all of [Fermat's] positive assertions with the one exception of the seemingly simple one which he made in his Last Theorem and which mathematicians, struggling for nearly 300 years, have been unable to prove: whenever Fermat asserted that he had ''proved'' anything, the statement, with the one exception noted, has subsequently been proved. Both his scrupulously honest character and his unrivalled penetration as an arithmetician substantiate the claim made for him by some, but not by all, that he knew what he was talking about when he asserted that he possessed a proof of his theorem."<ref> | + | This problem has confounded mathematicians for centuries, and there still is no proof for it using [[elementary techniques]]. [[Gauss]] and other mathematicians doubt that Fermat was able to prove it himself, but [[Cal Tech]] mathematics Professor E.T. Bell, who wrote the standard biography of all the great mathematicians, wryly observed that "the fox who could not get at the grapes declared they were sour." <ref>E.T. Bell, "Men of Mathematics" 72 (1937).</ref> "And so for all of [Fermat's] positive assertions with the one exception of the seemingly simple one which he made in his Last Theorem and which mathematicians, struggling for nearly 300 years, have been unable to prove: whenever Fermat asserted that he had ''proved'' anything, the statement, with the one exception noted, has subsequently been proved. Both his scrupulously honest character and his unrivalled penetration as an arithmetician substantiate the claim made for him by some, but not by all, that he knew what he was talking about when he asserted that he possessed a proof of his theorem."<ref>E.T. Bell, "Men of Mathematics" 71 (1937).</ref> |
The theorem is as follows: | The theorem is as follows: | ||
| Line 25: | Line 25: | ||
In the summer of 1986, Ken Ribet proved that Fermat's Last Theorem is a special case of the [[Taniyama–Shimura Conjecture]]. | In the summer of 1986, Ken Ribet proved that Fermat's Last Theorem is a special case of the [[Taniyama–Shimura Conjecture]]. | ||
| − | In a series of lectures in 1993, mathematician [[Andrew Wiles]] announced a proof using techniques in algebraic geometry, relying on the | + | == Claim of a Proof == |
| + | ===Andrew Wiles' Proposed Proof=== | ||
| + | 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 Is There a "Simple" Proof of Fermat's Last Theorem, Page 5 (occampress.com)]</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 - Is There a “Simple” Proof of Fermat's Last Theorem? Part (1) Introduction and Several New Approaches by Peter Schorer (cwru.edu) [http://www.cwru.edu/artsci/phil/Proving_FLT.pdf] —''copy this title and search online - a download is required for reading the first 3 pages of this paper, an online subscription is required to access all 89 pages''</ref> | ||
| + | |||
| + | ===The 1993 Proposed Proof Today=== | ||
| + | Unlike other mathematical breakthroughs, this claimed proof of 1993 has facilitated little, if any, insights or simplifications since then. Asking in recent years about the reliance by this proof on the disfavored [[Axiom of Choice]] has even resulted in [[censorship]] by some.<ref>https://meta.mathoverflow.net/questions/1336/why-was-this-question-about-axiom-of-choice-and-the-wiless-proof-of-flt-deleted</ref> | ||
| + | |||
| + | ===Transcendental Proof of a Finite Intellect?=== | ||
| + | |||
| + | A devout Catholic, Fermat held a judicial office within the [[Roman Catholic Church|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." | ||
| + | *[http://platonic-philosophy.org/files/Perl%20-%20The%20Good%20of%20the%20Intellect.pdf The Good of the Intellect, Eric D. Perl (platonic-philosophy.org)] pdf. "For Plotinus...the encounter with the Good demands a reaching beyond intellect, but is in no way opposed to or merely alongside intellect as an alternative to it."</ref> Pierre de Fermat's marginal notation may be an informal expression of his personal insight to the Theorem as being of itself "a truly marvelous demonstration of this proposition [of impossibility ''per se'']" directly relating to the wonder of this transcendent truth. | ||
| + | |||
| + | == See also == | ||
| + | *[[Paradox]] | ||
| + | *[[Pi]] | ||
| + | *[[Transcendental number]] | ||
| + | *[[Squaring the circle]] | ||
== References == | == References == | ||
Latest revision as of 21:25, July 12, 2019
Fermat's Last Theorem asserts that the well-known Pythagorean Theorem has no solutions for higher powers. That is,
has infinitely many solutions, such as {3, 4, 5} or {5, 12, 13}; but
has no solutions using positive whole numbers.
It was conjectured by the French mathematician Pierre de Fermat. He said he had proved this problem but that there was not enough room in the margin to state his proof:[1]
- "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:[2]
- "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."
This problem has confounded mathematicians for centuries, and there still is no proof for it using elementary techniques. Gauss and other mathematicians doubt that Fermat was able to prove it himself, but Cal Tech mathematics Professor E.T. Bell, who wrote the standard biography of all the great mathematicians, wryly observed that "the fox who could not get at the grapes declared they were sour." [3] "And so for all of [Fermat's] positive assertions with the one exception of the seemingly simple one which he made in his Last Theorem and which mathematicians, struggling for nearly 300 years, have been unable to prove: whenever Fermat asserted that he had proved anything, the statement, with the one exception noted, has subsequently been proved. Both his scrupulously honest character and his unrivalled penetration as an arithmetician substantiate the claim made for him by some, but not by all, that he knew what he was talking about when he asserted that he possessed a proof of his theorem."[4]
The theorem is as follows:
For integers n > 2, there are no nonzero integral solutions to: xn + yn = zn
In the summer of 1986, Ken Ribet proved that Fermat's Last Theorem is a special case of the Taniyama–Shimura Conjecture.
Contents
Claim of a Proof
Andrew Wiles' Proposed Proof
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.[5] 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.[5] Further criticism came from Marilyn vos Savant, known for her very high IQ and commentary on mathematics, in her column and book.[6][7] 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 ZFC axioms. It is an open question whether these tools can be formalized into a ZFC proof.[8]
The 1993 Proposed Proof Today
Unlike other mathematical breakthroughs, this claimed proof of 1993 has facilitated little, if any, insights or simplifications since then. Asking in recent years about the reliance by this proof on the disfavored Axiom of Choice has even resulted in censorship by some.[9]
Transcendental Proof of a Finite Intellect?
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.[10] Pierre de Fermat's marginal notation may be an informal expression of his personal insight to the Theorem as being of itself "a truly marvelous demonstration of this proposition [of impossibility per se]" directly relating to the wonder of this transcendent truth.
See also
References
- ↑ Nagell 1951, p. 252.
- ↑ Fermat's Last Theorem - Wolfram MathWorld (mathworld.wolfram.com)
- ↑ E.T. Bell, "Men of Mathematics" 72 (1937).
- ↑ E.T. Bell, "Men of Mathematics" 71 (1937).
- ↑ 5.0 5.1 Is There a "Simple" Proof of Fermat's Last Theorem, Page 5 (occampress.com)
- ↑ Ask Marilyn ® by Marilyn vos Savant, Parade Magazine. November 21, 1993
- ↑ 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
- ↑ Colin Mclarty - Is There a “Simple” Proof of Fermat's Last Theorem? Part (1) Introduction and Several New Approaches by Peter Schorer (cwru.edu) [1] —copy this title and search online - a download is required for reading the first 3 pages of this paper, an online subscription is required to access all 89 pages
- ↑ https://meta.mathoverflow.net/questions/1336/why-was-this-question-about-axiom-of-choice-and-the-wiless-proof-of-flt-deleted
- ↑ See the following:
- 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."
- 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."
- The Good of the Intellect, Eric D. Perl (platonic-philosophy.org) pdf. "For Plotinus...the encounter with the Good demands a reaching beyond intellect, but is in no way opposed to or merely alongside intellect as an alternative to it."
