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1.58 MB
Extraction Summary
8
People
3
Organizations
2
Locations
5
Events
2
Relationships
4
Quotes
Document Information
Type:
Book excerpt / scientific article (page 251)
File Size:
1.58 MB
Summary
This document appears to be a page (251) from a book or scientific article discussing mathematical history, specifically the solvability of Fermat's Last Theorem, Diophantine equations, and the Four Color Conjecture. It details the work of mathematicians like Yuri Matiyasevich, Julia Robinson, and Andrew Wiles, noting Wiles' secret work on Fermat's theorem at Princeton leading up to his 1995 announcement. The document bears a 'HOUSE_OVERSIGHT' Bates stamp, suggesting it was part of a larger production of documents to Congress, likely related to investigations into Jeffrey Epstein's connections with the scientific community.
People (8)
| Name | Role | Context |
|---|---|---|
| Yuri Matiyasevich | Mathematician |
Filled in the missing piece in Julia Robinson’s proof in 1970; used reduction method involving Turing machines.
|
| Julia Robinson | Mathematician |
Author of a proof related to Diophantine equations.
|
| Fermat | Mathematician (Historical) |
Referenced regarding Fermat's Last Theorem.
|
| Keijo Ruohonen | Researcher/Mathematician |
Demonstrated rewriting of Diophantine equations in 1972.
|
| Christoph Baxa | Researcher/Mathematician |
Demonstrated rewriting of Diophantine equations in 1993.
|
| J.P. Jones | Academic |
Of the University of Calgary; showed logic limit for regular Diophantine equations in 1993.
|
| Andrew Wiles | English Mathematics Professor |
Professor at Princeton; secretly worked on Fermat's Last Theorem; announced solution in 1995.
|
| Andrew Wiles' Wife | Spouse |
The only person Wiles told about his potential solution in late 1993.
|
Organizations (3)
| Name | Type | Context |
|---|---|---|
| University of Calgary |
Institution where J.P. Jones worked.
|
|
| Princeton |
University where Andrew Wiles was a professor.
|
|
| House Oversight Committee |
Implied by the Bates stamp 'HOUSE_OVERSIGHT'.
|
Timeline (5 events)
1970
Yuri Matiyasevich filled in the missing piece in Julia Robinson’s proof.
Unknown
1972
Keijo Ruohonen demonstrated Diophantine equations with exponential terms could be rewritten.
Unknown
1993
Christoph Baxa demonstrated Diophantine equations with exponential terms could be rewritten.
Unknown
1993
J.P. Jones showed the logic limit for regular Diophantine equations lies at thirteen unknowns.
University of Calgary
Relationships (2)
Matiyasevich filled in the missing piece in Julia Robinson’s proof.
Only told his wife late in 1993 when he suspected he might have a solution.
Key Quotes (4)
"Can humans solve ‘unsolvable’ problems?"Source
HOUSE_OVERSIGHT_015941.jpg
Quote #1
"Given an arbitrary map on a Euclidean plane, show the map can be colored in a maximum of four colors such that no adjacent area shares the same color."Source
HOUSE_OVERSIGHT_015941.jpg
Quote #2
"Finally, we have a proof that Fermat’s Last Theorem is unsolvable by a computer – or at least by a general purpose algorithm running on a computer."Source
HOUSE_OVERSIGHT_015941.jpg
Quote #3
"When I say secretly, he had not told anyone in his department, and only told his wife late in 1993 when he suspected he might have a solution."Source
HOUSE_OVERSIGHT_015941.jpg
Quote #4
Full Extracted Text
Complete text extracted from the document (2,387 characters)
Software
251
251
Can humans solve ‘unsolvable’ problems?
The question of whether Fermat’s Last Theorem could be solved mechanically remained unanswered until 1970 when Yuri Matiyasevich filled in the missing piece in Julia Robinson’s proof. Matiyasevich used an ingenious reduction method to match up sequences in Robinson’s theorem with a set of Turing machines. This showed that if Robinson’s theorem was false you could solve the halting problem and since you can’t solve the halting problem, then Robinson’s theorem must be true. All this effort proved Diophantine equations have no general algorithmic solution. This was a hugely important result but, as we noted earlier, Fermat’s Last Theorem is not, strictly speaking, a Diophantine. It is an exponential Diophantine equation. We still had no definitive answer to Fermat.
The question of whether Fermat’s Last Theorem could be solved mechanically remained unanswered until 1970 when Yuri Matiyasevich filled in the missing piece in Julia Robinson’s proof. Matiyasevich used an ingenious reduction method to match up sequences in Robinson’s theorem with a set of Turing machines. This showed that if Robinson’s theorem was false you could solve the halting problem and since you can’t solve the halting problem, then Robinson’s theorem must be true. All this effort proved Diophantine equations have no general algorithmic solution. This was a hugely important result but, as we noted earlier, Fermat’s Last Theorem is not, strictly speaking, a Diophantine. It is an exponential Diophantine equation. We still had no definitive answer to Fermat.
In 1972 Keijo Ruohonen and again in 1993, Christoph Baxa demonstrated that Diophantine equations with exponential terms could be rewritten as regular Diophantine equations with one additional complication – the necessity of adding an infinite set of terms to the end of the equation. In 1993, J.P. Jones of the University of Calgary showed the logic limit for regular Diophantine equations lies at thirteen unknowns. Matiyasevich had already pointed this out but never completed his proof. Since infinity is greater than thirteen, all exponential Diophantine equations are above the logic limit and, therefore, undecidable. Finally, we have a proof that Fermat’s Last Theorem is unsolvable by a computer – or at least by a general purpose algorithm running on a computer. Matiyasevich went on to show many mathematical problems can be rewritten as exponential Diophantine equations and that much of mathematics is undecidable. For example, the Four Color Conjecture:
“Given an arbitrary map on a Euclidean plane, show the map can be colored in a maximum of four colors such that no adjacent area shares the same color.”
Meanwhile, Andrew Wiles, an English mathematics Professor at Princeton had been secretly working on Fermat’s Last Theorem. When I say secretly, he had not told anyone in his department, and only told his wife late in 1993 when he suspected he might have a solution. He had been working on the problem a long time, having fallen in love with it at the age of 8! In 1995, after nearly 30 years work, he announced he
HOUSE_OVERSIGHT_015941
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