Euler's and Fermat's last theorems, the Simpsons and CDC6600
Mathologer・23 minutes read
Fermat's Last Theorem states that X^4 + Y^4 = Z^4 has no positive integer solutions, with Andrew Wiles proving it in 1993 for prime number exponents up to 125000. The proof involves showing a contradiction with two solutions, where the second one must be smaller, highlighting the complexity of Wiles's mathematical achievement.
Insights
- Fermat's Last Theorem asserts that no positive integer solutions exist for the equation X^4 + Y^4 = Z^4, with implications beyond this specific case.
- Euler's conjecture builds upon Fermat's Theorem, requiring multiple positive nth powers to sum up to another nth power, showcasing the interconnectedness of mathematical theories and the depth of analysis involved.
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What is Fermat's Last Theorem?
Fermat's Last Theorem states that the equation X^4 + Y^4 = Z^4 has no positive integer solutions.
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Summary
00:00
Mathematical Mysteries: Fermat, Euler, Pythagoras
- Fermat's Last Theorem states that the equation has no solutions in positive integers A, B, C for n greater than 2.
- Euler's conjecture extends Fermat's theorem to assert the impossibility of positive integer solutions for related equations.
- Fermat left a proof for the special case where the exponent is 4, covering multiple other cases.
- Mathematicians analyze Fermat's and Euler's equations to determine positive integer solutions.
- Pythagoras's theorem allows for positive integer solutions like 3, 4, 5 and 5, 12, 13.
- Babylonians knew about Pythagoras's theorem long before Pythagoras, with clay tablets showing their understanding.
- Euler's conjecture requires at least n positive nth powers to get a sum that is another nth power.
- Andrew Wiles announced a proof of Fermat's Last Theorem in 1993, a complex mathematical achievement.
- The Simpsons episode humorously presents a false counterexample to Fermat's Last Theorem.
- Divisibility by 4 can be used to disprove equations like the ones presented in The Simpsons.
15:06
"Mathematical Mysteries and Supercomputers"
- The CDC6600 was the first supercomputer developed.
- The CDC6600 and its successors only managed to disprove one instance of Euler's conjecture.
- A mathematical consultant for the Simpsons may resort to prank solutions if faced with a looming deadline.
- Mathematicians may use a proof by contradiction strategy to show the absence of a solution to an equation.
- Fermat's Last Theorem states that X^4 + Y^4 = Z^4 has no positive integer solutions.
- Euler proved Fermat's Last Theorem for the exponent 3, which also covers multiples of this exponent.
- Wiles' proof of Fermat's Last Theorem covered prime number exponents up to 125000.
- After dividing by 4, any even power of an odd number must have a remainder of 1.
- In a three-term equation, if two terms share a common factor, the third term must also have that factor.
- Prime factorization of a fourth power requires all individual prime powers to be fourth powers themselves.
30:19
Fermat's Last Theorem: Two Solutions, One Contradiction
- The proof of Fermat's Last Theorem involves two solutions, with the second solution needing to be smaller than the first, leading to a contradiction that completes the proof. This simple explanation hints at the complexity of Andrew Wiles's proof, which is significantly more intricate.
