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Introduction Sketch of Proof Going Into the Details Hilberts Tenth Problem John Lindsay Orr Department of Mathematics Univesity of NebraskaLincoln September 15, 2005 John Lindsay Orr Hilberts Tenth Problem Introduction Sketch of

  1. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations How a Turing Machine Works At each step the machine: scans the current cell while in state x 1 reads the value ( y ) from that cell 2 writes a value W ( x , y ) to the cell 3 moves in direction D ( x , y ) 4 enters state S ( x , y ) 5 So the machine is determined by three finite functions: W : X × Y − → Y , D : X × Y − → {− 1 , 0 , 1 } , and S : X × Y − → X The machine also has a single initial state x 1 and some final states . John Lindsay Orr Hilbert’s Tenth Problem

  2. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations How a Turing Machine Works At each step the machine: scans the current cell while in state x 1 reads the value ( y ) from that cell 2 writes a value W ( x , y ) to the cell 3 moves in direction D ( x , y ) 4 enters state S ( x , y ) 5 So the machine is determined by three finite functions: W : X × Y − → Y , D : X × Y − → {− 1 , 0 , 1 } , and S : X × Y − → X The machine also has a single initial state x 1 and some final states . John Lindsay Orr Hilbert’s Tenth Problem

  3. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations How a Turing Machine Works At each step the machine: scans the current cell while in state x 1 reads the value ( y ) from that cell 2 writes a value W ( x , y ) to the cell 3 moves in direction D ( x , y ) 4 enters state S ( x , y ) 5 So the machine is determined by three finite functions: W : X × Y − → Y , D : X × Y − → {− 1 , 0 , 1 } , and S : X × Y − → X The machine also has a single initial state x 1 and some final states . John Lindsay Orr Hilbert’s Tenth Problem

  4. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations How a Turing Machine Works At each step the machine: scans the current cell while in state x 1 reads the value ( y ) from that cell 2 writes a value W ( x , y ) to the cell 3 moves in direction D ( x , y ) 4 enters state S ( x , y ) 5 So the machine is determined by three finite functions: W : X × Y − → Y , D : X × Y − → {− 1 , 0 , 1 } , and S : X × Y − → X The machine also has a single initial state x 1 and some final states . John Lindsay Orr Hilbert’s Tenth Problem

  5. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations How a Turing Machine Works At each step the machine: scans the current cell while in state x 1 reads the value ( y ) from that cell 2 writes a value W ( x , y ) to the cell 3 moves in direction D ( x , y ) 4 enters state S ( x , y ) 5 So the machine is determined by three finite functions: W : X × Y − → Y , D : X × Y − → {− 1 , 0 , 1 } , and S : X × Y − → X The machine also has a single initial state x 1 and some final states . John Lindsay Orr Hilbert’s Tenth Problem

  6. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations How a Turing Machine Works At each step the machine: scans the current cell while in state x 1 reads the value ( y ) from that cell 2 writes a value W ( x , y ) to the cell 3 moves in direction D ( x , y ) 4 enters state S ( x , y ) 5 So the machine is determined by three finite functions: W : X × Y − → Y , D : X × Y − → {− 1 , 0 , 1 } , and S : X × Y − → X The machine also has a single initial state x 1 and some final states . John Lindsay Orr Hilbert’s Tenth Problem

  7. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations How a Turing Machine Works At each step the machine: scans the current cell while in state x 1 reads the value ( y ) from that cell 2 writes a value W ( x , y ) to the cell 3 moves in direction D ( x , y ) 4 enters state S ( x , y ) 5 So the machine is determined by three finite functions: W : X × Y − → Y , D : X × Y − → {− 1 , 0 , 1 } , and S : X × Y − → X The machine also has a single initial state x 1 and some final states . John Lindsay Orr Hilbert’s Tenth Problem

  8. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations How a Turing Machine Works At each step the machine: scans the current cell while in state x 1 reads the value ( y ) from that cell 2 writes a value W ( x , y ) to the cell 3 moves in direction D ( x , y ) 4 enters state S ( x , y ) 5 So the machine is determined by three finite functions: W : X × Y − → Y , D : X × Y − → {− 1 , 0 , 1 } , and S : X × Y − → X The machine also has a single initial state x 1 and some final states . John Lindsay Orr Hilbert’s Tenth Problem

  9. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations How to Program a Turing Machine Build simple machines that do basic operations, like: LEFT or RIGHT WRITE ( y ) READ ( y ) STOP or NEVERSTOP Learn how to compose machines: if ( M 1 ) { M 2 } or while ( M 1 ) { M 2 } John Lindsay Orr Hilbert’s Tenth Problem

  10. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations How to Program a Turing Machine Build simple machines that do basic operations, like: LEFT or RIGHT WRITE ( y ) READ ( y ) STOP or NEVERSTOP Learn how to compose machines: if ( M 1 ) { M 2 } or while ( M 1 ) { M 2 } John Lindsay Orr Hilbert’s Tenth Problem

  11. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations How to Program a Turing Machine Build simple machines that do basic operations, like: LEFT or RIGHT WRITE ( y ) READ ( y ) STOP or NEVERSTOP Learn how to compose machines: if ( M 1 ) { M 2 } or while ( M 1 ) { M 2 } John Lindsay Orr Hilbert’s Tenth Problem

  12. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations How to Program a Turing Machine Build simple machines that do basic operations, like: LEFT or RIGHT WRITE ( y ) READ ( y ) STOP or NEVERSTOP Learn how to compose machines: if ( M 1 ) { M 2 } or while ( M 1 ) { M 2 } John Lindsay Orr Hilbert’s Tenth Problem

  13. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations How to Program a Turing Machine Build simple machines that do basic operations, like: LEFT or RIGHT WRITE ( y ) READ ( y ) STOP or NEVERSTOP Learn how to compose machines: if ( M 1 ) { M 2 } or while ( M 1 ) { M 2 } John Lindsay Orr Hilbert’s Tenth Problem

  14. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations How to Program a Turing Machine Build simple machines that do basic operations, like: LEFT or RIGHT WRITE ( y ) READ ( y ) STOP or NEVERSTOP Learn how to compose machines: if ( M 1 ) { M 2 } or while ( M 1 ) { M 2 } John Lindsay Orr Hilbert’s Tenth Problem

  15. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Say that a set S ⊆ N is Turing decidable if there is a Turing machine M such that, whenever M is started with initial data on the tape encoding a n ∈ N : M halts in state q 2 if n ∈ S M halts in state q 3 if n �∈ S John Lindsay Orr Hilbert’s Tenth Problem

  16. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations How to Answer Hilbert’s Tenth Problem Imagine indexing all possible diophantine equations in some order. E.g. D 1 , D 2 , D 3 , . . . . Let S = { k : D k has a solution } . Hilbert’s 10th problem becomes: Question Is S Turing decidable? John Lindsay Orr Hilbert’s Tenth Problem

  17. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Say that a set S ⊆ N is Turing semidecidable if there is a Turing machine M such that, whenever M is started with initial data on the tape encoding a n ∈ N : if n ∈ S then M eventually halts if n �∈ S then M never halts John Lindsay Orr Hilbert’s Tenth Problem

  18. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Lemma If S is Turing decidable then S and S c are Turing semidecidable. Proof. Let M be a machine that decides S . To semidecide S use the machine: if ( M ) { STOP }; NEVERSTOP To semidecide S c use the machine: if ( M ) { NEVERSTOP } STOP ; John Lindsay Orr Hilbert’s Tenth Problem

  19. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Lemma If S is Turing decidable then S and S c are Turing semidecidable. Proof. Let M be a machine that decides S . To semidecide S use the machine: if ( M ) { STOP }; NEVERSTOP To semidecide S c use the machine: if ( M ) { NEVERSTOP } STOP ; John Lindsay Orr Hilbert’s Tenth Problem

  20. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Lemma If S is Turing decidable then S and S c are Turing semidecidable. Proof. Let M be a machine that decides S . To semidecide S use the machine: if ( M ) { STOP }; NEVERSTOP To semidecide S c use the machine: if ( M ) { NEVERSTOP } STOP ; John Lindsay Orr Hilbert’s Tenth Problem

  21. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Theorem The set S is Turing decidable if and only if S and S c are Turing semidecidable. John Lindsay Orr Hilbert’s Tenth Problem

  22. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Outline Introduction 1 Disclaimer History and Statement of the Problem Sketch of Proof 2 Turing Machines and Decidability Diophantine Sets Universal Diophantine Equations Going Into the Details 3 Working with Diophantine Sets Coding n -tuples John Lindsay Orr Hilbert’s Tenth Problem

  23. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Definition Say that a set S ⊆ N k is diophantine if there exists a diophantine equation D ( a 1 , . . . , a k , x 1 , . . . , x n ) = 0 such that ( a 1 , . . . , a k ) ∈ S if and only if D ( a 1 , . . . , a k , x 1 , . . . , x n ) = 0 has a solution in N n . John Lindsay Orr Hilbert’s Tenth Problem

  24. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Definition Say that a set S ⊆ N k is diophantine if there exists a diophantine equation D ( a 1 , . . . , a k , x 1 , . . . , x n ) = 0 such that ( a 1 , . . . , a k ) ∈ S if and only if D ( a 1 , . . . , a k , x 1 , . . . , x n ) = 0 has a solution in N n . Example. The set { ( a , b ) : gcd ( a , b ) = 1 } is diophantine. John Lindsay Orr Hilbert’s Tenth Problem

  25. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Definition Say that a set S ⊆ N k is diophantine if there exists a diophantine equation D ( a 1 , . . . , a k , x 1 , . . . , x n ) = 0 such that ( a 1 , . . . , a k ) ∈ S if and only if D ( a 1 , . . . , a k , x 1 , . . . , x n ) = 0 has a solution in N n . Example. The set { ( a , b ) : gcd ( a , b ) = 1 } is diophantine. (Take D ( a , b , x , y ) = ax − by − 1.) John Lindsay Orr Hilbert’s Tenth Problem

  26. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Example. The set { a : a is not a prime } is diophantine. Proof. Let D ( a , x , y ) = ( x + 2 )( y + 2 ) − a John Lindsay Orr Hilbert’s Tenth Problem

  27. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations In fact, the set { a : a is a prime } is diophantine. Factoid. A set S ⊆ N is diophantine if and only if S is the set of non-negative values taken by some integer-coefficient polynomial as its variables range over N . Thus, incredibly, { prime numbers } = N ∩ { D ( x 1 , . . . , x n : x 1 , . . . , x n ∈ N } John Lindsay Orr Hilbert’s Tenth Problem

  28. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Lemma Every diophantine set is Turing semidecidable. John Lindsay Orr Hilbert’s Tenth Problem

  29. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Proof. S has a diophantine representation D ( a 1 , . . . , a k , x 1 , . . . , x n ) = 0 Initialize the tape with ( a 1 , . . . , a k ) ∈ N k , and run: foreach x = ( x 1 , . . . , x n ) ∈ N n { if ( D ( a 1 , . . . , a k , x 1 , . . . , x n ) = 0 ) { STOP } } John Lindsay Orr Hilbert’s Tenth Problem

  30. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Theorem Every Turing semidecidable set is diophantine. Corollary A set is diophantine ⇐ ⇒ it is Turing semidecidable. John Lindsay Orr Hilbert’s Tenth Problem

  31. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Theorem Every Turing semidecidable set is diophantine. Corollary A set is diophantine ⇐ ⇒ it is Turing semidecidable. John Lindsay Orr Hilbert’s Tenth Problem

  32. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Summary What have we learned? S is decidable ⇒ S , S c are semidecidable ⇐ ⇒ S , S c are diophantine ⇐ So one way to show a set is not decidable is to show that one of S or S c is not diophantine. John Lindsay Orr Hilbert’s Tenth Problem

  33. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Outline Introduction 1 Disclaimer History and Statement of the Problem Sketch of Proof 2 Turing Machines and Decidability Diophantine Sets Universal Diophantine Equations Going Into the Details 3 Working with Diophantine Sets Coding n -tuples John Lindsay Orr Hilbert’s Tenth Problem

  34. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Definition The integer-coefficient polynomial U ( a 1 , . . . , a k , c , y 1 , . . . , y w ) is a universal diophantine polynomial if, for any diophantine equation D ( a 1 , . . . , a k , x 1 , . . . , x n ) = 0 we can find a code c ∈ N such that ∃ x 1 , . . . , x n with D ( a 1 , . . . , a k , x 1 , . . . , x n ) = 0 ⇐ ⇒ ∃ y 1 , . . . , y w with U ( a 1 , . . . , a k , c , y 1 , . . . , y w ) = 0 John Lindsay Orr Hilbert’s Tenth Problem

  35. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Theorem For each k, there exists a universal diophantine equation U k ( a 1 , . . . , a k , c , y 1 , . . . , y w ) Let H 0 = { c : U 0 ( c , y 1 , . . . , y v ) = 0 has a solution } This is our “enumeration of the solvable diophantine equations”. We shall show that H 0 is diophantine and H c 0 is not! John Lindsay Orr Hilbert’s Tenth Problem

  36. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Theorem For each k, there exists a universal diophantine equation U k ( a 1 , . . . , a k , c , y 1 , . . . , y w ) Let H 0 = { c : U 0 ( c , y 1 , . . . , y v ) = 0 has a solution } This is our “enumeration of the solvable diophantine equations”. We shall show that H 0 is diophantine and H c 0 is not! John Lindsay Orr Hilbert’s Tenth Problem

  37. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Theorem For each k, there exists a universal diophantine equation U k ( a 1 , . . . , a k , c , y 1 , . . . , y w ) Let H 0 = { c : U 0 ( c , y 1 , . . . , y v ) = 0 has a solution } This is our “enumeration of the solvable diophantine equations”. We shall show that H 0 is diophantine and H c 0 is not! John Lindsay Orr Hilbert’s Tenth Problem

  38. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Let H 1 = { k : U 1 ( k , k , y 1 , . . . , y w ) = 0 has a solution } Claim. H 1 is a diophantine set but H c 1 is not. Proof. (First part) Write D ( k , y 1 , . . . , y w ) = U 1 ( k , k , y 1 , . . . , y w ) . Then k ∈ H 1 ⇐ ⇒ D ( k , y 1 , . . . , y w ) has a solution Thus H 1 is diophantine. John Lindsay Orr Hilbert’s Tenth Problem

  39. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Let H 1 = { k : U 1 ( k , k , y 1 , . . . , y w ) = 0 has a solution } Claim. H 1 is a diophantine set but H c 1 is not. Proof. (First part) Write D ( k , y 1 , . . . , y w ) = U 1 ( k , k , y 1 , . . . , y w ) . Then k ∈ H 1 ⇐ ⇒ D ( k , y 1 , . . . , y w ) has a solution Thus H 1 is diophantine. John Lindsay Orr Hilbert’s Tenth Problem

  40. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Let H 1 = { k : U 1 ( k , k , y 1 , . . . , y w ) = 0 has a solution } Claim. H 1 is a diophantine set but H c 1 is not. Proof. (First part) Write D ( k , y 1 , . . . , y w ) = U 1 ( k , k , y 1 , . . . , y w ) . Then k ∈ H 1 ⇐ ⇒ D ( k , y 1 , . . . , y w ) has a solution Thus H 1 is diophantine. John Lindsay Orr Hilbert’s Tenth Problem

  41. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Let H 1 = { k : U 1 ( k , k , y 1 , . . . , y w ) = 0 has a solution } Claim. H 1 is a diophantine set but H c 1 is not. Proof. (First part) Write D ( k , y 1 , . . . , y w ) = U 1 ( k , k , y 1 , . . . , y w ) . Then k ∈ H 1 ⇐ ⇒ D ( k , y 1 , . . . , y w ) has a solution Thus H 1 is diophantine. John Lindsay Orr Hilbert’s Tenth Problem

  42. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Let H 1 = { k : U 1 ( k , k , y 1 , . . . , y w ) = 0 has a solution } Claim. H 1 is a diophantine set but H c 1 is not. Proof. (Second part) If H c 1 were diophantine there would be a code, k , for it. But then ask: Does U 1 ( k , k , y 1 , . . . , y w ) = 0 have a solution? If “yes” then k ∈ H 1 . But k is the code for the set H c 1 so in general: ⇒ a ∈ H c U 1 ( k , k , y 1 , . . . , y w ) = 0 has a solution ⇐ 1 Thus, k ∈ H c 1 . Contradiction! If “no” then k ∈ H c 1 . But likewise a �∈ H c 1 . Contradiction! John Lindsay Orr Hilbert’s Tenth Problem

  43. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Let H 1 = { k : U 1 ( k , k , y 1 , . . . , y w ) = 0 has a solution } Claim. H 1 is a diophantine set but H c 1 is not. Proof. (Second part) If H c 1 were diophantine there would be a code, k , for it. But then ask: Does U 1 ( k , k , y 1 , . . . , y w ) = 0 have a solution? If “yes” then k ∈ H 1 . But k is the code for the set H c 1 so in general: ⇒ a ∈ H c U 1 ( k , k , y 1 , . . . , y w ) = 0 has a solution ⇐ 1 Thus, k ∈ H c 1 . Contradiction! If “no” then k ∈ H c 1 . But likewise a �∈ H c 1 . Contradiction! John Lindsay Orr Hilbert’s Tenth Problem

  44. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Let H 1 = { k : U 1 ( k , k , y 1 , . . . , y w ) = 0 has a solution } Claim. H 1 is a diophantine set but H c 1 is not. Proof. (Second part) If H c 1 were diophantine there would be a code, k , for it. But then ask: Does U 1 ( k , k , y 1 , . . . , y w ) = 0 have a solution? If “yes” then k ∈ H 1 . But k is the code for the set H c 1 so in general: ⇒ a ∈ H c U 1 ( k , k , y 1 , . . . , y w ) = 0 has a solution ⇐ 1 Thus, k ∈ H c 1 . Contradiction! If “no” then k ∈ H c 1 . But likewise a �∈ H c 1 . Contradiction! John Lindsay Orr Hilbert’s Tenth Problem

  45. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Let H 1 = { k : U 1 ( k , k , y 1 , . . . , y w ) = 0 has a solution } Claim. H 1 is a diophantine set but H c 1 is not. Proof. (Second part) If H c 1 were diophantine there would be a code, k , for it. But then ask: Does U 1 ( k , k , y 1 , . . . , y w ) = 0 have a solution? If “yes” then k ∈ H 1 . But k is the code for the set H c 1 so in general: ⇒ a ∈ H c U 1 ( k , k , y 1 , . . . , y w ) = 0 has a solution ⇐ 1 Thus, k ∈ H c 1 . Contradiction! If “no” then k ∈ H c 1 . But likewise a �∈ H c 1 . Contradiction! John Lindsay Orr Hilbert’s Tenth Problem

  46. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Let H 1 = { k : U 1 ( k , k , y 1 , . . . , y w ) = 0 has a solution } Claim. H 1 is a diophantine set but H c 1 is not. Proof. (Second part) If H c 1 were diophantine there would be a code, k , for it. But then ask: Does U 1 ( k , k , y 1 , . . . , y w ) = 0 have a solution? If “yes” then k ∈ H 1 . But k is the code for the set H c 1 so in general: ⇒ a ∈ H c U 1 ( k , k , y 1 , . . . , y w ) = 0 has a solution ⇐ 1 Thus, k ∈ H c 1 . Contradiction! If “no” then k ∈ H c 1 . But likewise a �∈ H c 1 . Contradiction! John Lindsay Orr Hilbert’s Tenth Problem

  47. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Let H 1 = { k : U 1 ( k , k , y 1 , . . . , y w ) = 0 has a solution } Claim. H 1 is a diophantine set but H c 1 is not. Proof. (Second part) If H c 1 were diophantine there would be a code, k , for it. But then ask: Does U 1 ( k , k , y 1 , . . . , y w ) = 0 have a solution? If “yes” then k ∈ H 1 . But k is the code for the set H c 1 so in general: ⇒ a ∈ H c U 1 ( k , k , y 1 , . . . , y w ) = 0 has a solution ⇐ 1 Thus, k ∈ H c 1 . Contradiction! If “no” then k ∈ H c 1 . But likewise a �∈ H c 1 . Contradiction! John Lindsay Orr Hilbert’s Tenth Problem

  48. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Let H 0 = { k : U 0 ( k , y 1 , . . . , y v ) = 0 has a solution } U 1 ( a , k , y 1 , . . . , y w ) = 0 has a solution ⇐ ⇒ D k ( a , x 1 , . . . , x n ) = 0 has a solution ⇐ ⇒ W ( x 1 , . . . , x n ) = 0 has a solution ⇐ ⇒ U 0 ( c ( a , k ) , y 1 , . . . , y v ) = 0 has a solution Thus c ( k , k ) ∈ H 0 ⇐ ⇒ k ∈ H 1 c ( k , k ) ∈ H c ⇒ k ∈ H c 0 ⇐ 1 Fact. c ( a , k ) is a diophantine polynomial ⇒ H c 0 is not diophantine! John Lindsay Orr Hilbert’s Tenth Problem

  49. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Let H 0 = { k : U 0 ( k , y 1 , . . . , y v ) = 0 has a solution } U 1 ( a , k , y 1 , . . . , y w ) = 0 has a solution ⇐ ⇒ D k ( a , x 1 , . . . , x n ) = 0 has a solution ⇐ ⇒ W ( x 1 , . . . , x n ) = 0 has a solution ⇐ ⇒ U 0 ( c ( a , k ) , y 1 , . . . , y v ) = 0 has a solution Thus c ( k , k ) ∈ H 0 ⇐ ⇒ k ∈ H 1 c ( k , k ) ∈ H c ⇒ k ∈ H c 0 ⇐ 1 Fact. c ( a , k ) is a diophantine polynomial ⇒ H c 0 is not diophantine! John Lindsay Orr Hilbert’s Tenth Problem

  50. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Let H 0 = { k : U 0 ( k , y 1 , . . . , y v ) = 0 has a solution } U 1 ( a , k , y 1 , . . . , y w ) = 0 has a solution ⇐ ⇒ D k ( a , x 1 , . . . , x n ) = 0 has a solution ⇐ ⇒ W ( x 1 , . . . , x n ) = 0 has a solution ⇐ ⇒ U 0 ( c ( a , k ) , y 1 , . . . , y v ) = 0 has a solution Thus c ( k , k ) ∈ H 0 ⇐ ⇒ k ∈ H 1 c ( k , k ) ∈ H c ⇒ k ∈ H c 0 ⇐ 1 Fact. c ( a , k ) is a diophantine polynomial ⇒ H c 0 is not diophantine! John Lindsay Orr Hilbert’s Tenth Problem

  51. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Let H 0 = { k : U 0 ( k , y 1 , . . . , y v ) = 0 has a solution } U 1 ( a , k , y 1 , . . . , y w ) = 0 has a solution ⇐ ⇒ D k ( a , x 1 , . . . , x n ) = 0 has a solution ⇐ ⇒ W ( x 1 , . . . , x n ) = 0 has a solution ⇐ ⇒ U 0 ( c ( a , k ) , y 1 , . . . , y v ) = 0 has a solution Thus c ( k , k ) ∈ H 0 ⇐ ⇒ k ∈ H 1 c ( k , k ) ∈ H c ⇒ k ∈ H c 0 ⇐ 1 Fact. c ( a , k ) is a diophantine polynomial ⇒ H c 0 is not diophantine! John Lindsay Orr Hilbert’s Tenth Problem

  52. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Let H 0 = { k : U 0 ( k , y 1 , . . . , y v ) = 0 has a solution } U 1 ( a , k , y 1 , . . . , y w ) = 0 has a solution ⇐ ⇒ D k ( a , x 1 , . . . , x n ) = 0 has a solution ⇐ ⇒ W ( x 1 , . . . , x n ) = 0 has a solution ⇐ ⇒ U 0 ( c ( a , k ) , y 1 , . . . , y v ) = 0 has a solution Thus c ( k , k ) ∈ H 0 ⇐ ⇒ k ∈ H 1 c ( k , k ) ∈ H c ⇒ k ∈ H c 0 ⇐ 1 Fact. c ( a , k ) is a diophantine polynomial ⇒ H c 0 is not diophantine! John Lindsay Orr Hilbert’s Tenth Problem

  53. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Let H 0 = { k : U 0 ( k , y 1 , . . . , y v ) = 0 has a solution } U 1 ( a , k , y 1 , . . . , y w ) = 0 has a solution ⇐ ⇒ D k ( a , x 1 , . . . , x n ) = 0 has a solution ⇐ ⇒ W ( x 1 , . . . , x n ) = 0 has a solution ⇐ ⇒ U 0 ( c ( a , k ) , y 1 , . . . , y v ) = 0 has a solution Thus c ( k , k ) ∈ H 0 ⇐ ⇒ k ∈ H 1 c ( k , k ) ∈ H c ⇒ k ∈ H c 0 ⇐ 1 Fact. c ( a , k ) is a diophantine polynomial ⇒ H c 0 is not diophantine! John Lindsay Orr Hilbert’s Tenth Problem

  54. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Let H 0 = { k : U 0 ( k , y 1 , . . . , y v ) = 0 has a solution } U 1 ( a , k , y 1 , . . . , y w ) = 0 has a solution ⇐ ⇒ D k ( a , x 1 , . . . , x n ) = 0 has a solution ⇐ ⇒ W ( x 1 , . . . , x n ) = 0 has a solution ⇐ ⇒ U 0 ( c ( a , k ) , y 1 , . . . , y v ) = 0 has a solution Thus c ( k , k ) ∈ H 0 ⇐ ⇒ k ∈ H 1 c ( k , k ) ∈ H c ⇒ k ∈ H c 0 ⇐ 1 Fact. c ( a , k ) is a diophantine polynomial ⇒ H c 0 is not diophantine! John Lindsay Orr Hilbert’s Tenth Problem

  55. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Let H 0 = { k : U 0 ( k , y 1 , . . . , y v ) = 0 has a solution } U 1 ( a , k , y 1 , . . . , y w ) = 0 has a solution ⇐ ⇒ D k ( a , x 1 , . . . , x n ) = 0 has a solution ⇐ ⇒ W ( x 1 , . . . , x n ) = 0 has a solution ⇐ ⇒ U 0 ( c ( a , k ) , y 1 , . . . , y v ) = 0 has a solution Thus c ( k , k ) ∈ H 0 ⇐ ⇒ k ∈ H 1 c ( k , k ) ∈ H c ⇒ k ∈ H c 0 ⇐ 1 Fact. c ( a , k ) is a diophantine polynomial ⇒ H c 0 is not diophantine! John Lindsay Orr Hilbert’s Tenth Problem

  56. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Summary We have seen that H 0 = { k : U 0 ( k , y 1 , . . . , y v ) = 0 has a solution } is not Turing decidable. The elements of H 0 are in one-to-one correspondence with the solvable diophantine equations. Thus, there is no algorithm to decide which diophantine equations are solvable and which are not. John Lindsay Orr Hilbert’s Tenth Problem

  57. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Summary We have seen that H 0 = { k : U 0 ( k , y 1 , . . . , y v ) = 0 has a solution } is not Turing decidable. The elements of H 0 are in one-to-one correspondence with the solvable diophantine equations. Thus, there is no algorithm to decide which diophantine equations are solvable and which are not. John Lindsay Orr Hilbert’s Tenth Problem

  58. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Summary We have seen that H 0 = { k : U 0 ( k , y 1 , . . . , y v ) = 0 has a solution } is not Turing decidable. The elements of H 0 are in one-to-one correspondence with the solvable diophantine equations. Thus, there is no algorithm to decide which diophantine equations are solvable and which are not. John Lindsay Orr Hilbert’s Tenth Problem

  59. Introduction Turing Machines and Decidability Sketch of Proof Diophantine Sets Going Into the Details Universal Diophantine Equations Summary We have seen that H 0 = { k : U 0 ( k , y 1 , . . . , y v ) = 0 has a solution } is not Turing decidable. The elements of H 0 are in one-to-one correspondence with the solvable diophantine equations. Thus, there is no algorithm to decide which diophantine equations are solvable and which are not. John Lindsay Orr Hilbert’s Tenth Problem

  60. Introduction Working with Diophantine Sets Sketch of Proof Coding n -tuples Going Into the Details Outline Introduction 1 Disclaimer History and Statement of the Problem Sketch of Proof 2 Turing Machines and Decidability Diophantine Sets Universal Diophantine Equations Going Into the Details 3 Working with Diophantine Sets Coding n -tuples John Lindsay Orr Hilbert’s Tenth Problem

  61. Introduction Working with Diophantine Sets Sketch of Proof Coding n -tuples Going Into the Details Unions and Intersections Let S 1 , S 2 ⊆ N k be diophantine sets with representations ( a 1 , . . . , a k ) ∈ S 1 ⇐ ⇒ D 1 ( a 1 , . . . , a k , x 1 , . . . , x m ) = 0 has a solution and ( a 1 , . . . , a k ) ∈ S 1 ⇐ ⇒ D 2 ( a 1 , . . . , a k , y 1 , . . . , y n ) = 0 has a solution Then S 1 ∪ S 2 and S 1 ∩ S 2 are diophantine sets. Proof. Consider D 1 ( a 1 , . . . , a k , x 1 , . . . , x m ) D 2 ( a 1 , . . . , a k , y 1 , . . . , y n ) = 0 and D 1 ( a 1 , . . . , a k , x 1 , . . . , x m ) 2 + D 2 ( a 1 , . . . , a k , y 1 , . . . , y n ) 2 = 0 John Lindsay Orr Hilbert’s Tenth Problem

  62. Introduction Working with Diophantine Sets Sketch of Proof Coding n -tuples Going Into the Details Unions and Intersections Let S 1 , S 2 ⊆ N k be diophantine sets with representations ( a 1 , . . . , a k ) ∈ S 1 ⇐ ⇒ D 1 ( a 1 , . . . , a k , x 1 , . . . , x m ) = 0 has a solution and ( a 1 , . . . , a k ) ∈ S 1 ⇐ ⇒ D 2 ( a 1 , . . . , a k , y 1 , . . . , y n ) = 0 has a solution Then S 1 ∪ S 2 and S 1 ∩ S 2 are diophantine sets. Proof. Consider D 1 ( a 1 , . . . , a k , x 1 , . . . , x m ) D 2 ( a 1 , . . . , a k , y 1 , . . . , y n ) = 0 and D 1 ( a 1 , . . . , a k , x 1 , . . . , x m ) 2 + D 2 ( a 1 , . . . , a k , y 1 , . . . , y n ) 2 = 0 John Lindsay Orr Hilbert’s Tenth Problem

  63. Introduction Working with Diophantine Sets Sketch of Proof Coding n -tuples Going Into the Details Some Basic Diophantine Sets The set { ( a , b ) : aRb } is diophantine when “ R ” is one of the relations: a = b (consider “ ∃ x s.t. x + ( a − b ) 2 = 0”) a < b (consider “ ∃ x s.t. a + x + 1 = b ”) a | b (consider “ ∃ x s.t. ax = b ”) John Lindsay Orr Hilbert’s Tenth Problem

  64. Introduction Working with Diophantine Sets Sketch of Proof Coding n -tuples Going Into the Details Some Basic Diophantine Sets The set { ( a , b ) : aRb } is diophantine when “ R ” is one of the relations: a = b (consider “ ∃ x s.t. x + ( a − b ) 2 = 0”) a < b (consider “ ∃ x s.t. a + x + 1 = b ”) a | b (consider “ ∃ x s.t. ax = b ”) John Lindsay Orr Hilbert’s Tenth Problem

  65. Introduction Working with Diophantine Sets Sketch of Proof Coding n -tuples Going Into the Details Some Basic Diophantine Sets The set { ( a , b ) : aRb } is diophantine when “ R ” is one of the relations: a = b (consider “ ∃ x s.t. x + ( a − b ) 2 = 0”) a < b (consider “ ∃ x s.t. a + x + 1 = b ”) a | b (consider “ ∃ x s.t. ax = b ”) John Lindsay Orr Hilbert’s Tenth Problem

  66. Introduction Working with Diophantine Sets Sketch of Proof Coding n -tuples Going Into the Details Some Basic Diophantine Sets The set { ( a , b ) : aRb } is diophantine when “ R ” is one of the relations: a = b (consider “ ∃ x s.t. x + ( a − b ) 2 = 0”) a < b (consider “ ∃ x s.t. a + x + 1 = b ”) a | b (consider “ ∃ x s.t. ax = b ”) John Lindsay Orr Hilbert’s Tenth Problem

  67. Introduction Working with Diophantine Sets Sketch of Proof Coding n -tuples Going Into the Details Some Basic Diophantine Sets The set { ( a , b ) : aRb } is diophantine when “ R ” is one of the relations: a = b (consider “ ∃ x s.t. x + ( a − b ) 2 = 0”) a < b (consider “ ∃ x s.t. a + x + 1 = b ”) a | b (consider “ ∃ x s.t. ax = b ”) The set { ( a , b , c ) : a = rem ( b , c ) } is diophantine. Proof. a = rem ( b , c ) ⇐ ⇒ a < c & c | b − a ⇒∃ x , y s.t. ( a + x + 1 − b ) 2 + ( cy − ( b − a )) 2 = 0 ⇐ John Lindsay Orr Hilbert’s Tenth Problem

  68. Introduction Working with Diophantine Sets Sketch of Proof Coding n -tuples Going Into the Details Some Basic Diophantine Sets The set { ( a , b ) : aRb } is diophantine when “ R ” is one of the relations: a = b (consider “ ∃ x s.t. x + ( a − b ) 2 = 0”) a < b (consider “ ∃ x s.t. a + x + 1 = b ”) a | b (consider “ ∃ x s.t. ax = b ”) The set { ( a , b , c ) : a = rem ( b , c ) } is diophantine. Proof. a = rem ( b , c ) ⇐ ⇒ a < c & c | b − a ⇒∃ x , y s.t. ( a + x + 1 − b ) 2 + ( cy − ( b − a )) 2 = 0 ⇐ John Lindsay Orr Hilbert’s Tenth Problem

  69. Introduction Working with Diophantine Sets Sketch of Proof Coding n -tuples Going Into the Details Some Basic Diophantine Sets The set { ( a , b ) : aRb } is diophantine when “ R ” is one of the relations: a = b (consider “ ∃ x s.t. x + ( a − b ) 2 = 0”) a < b (consider “ ∃ x s.t. a + x + 1 = b ”) a | b (consider “ ∃ x s.t. ax = b ”) The set { ( a , b , c ) : a = rem ( b , c ) } is diophantine. Proof. a = rem ( b , c ) ⇐ ⇒ a < c & c | b − a ⇒∃ x , y s.t. ( a + x + 1 − b ) 2 + ( cy − ( b − a )) 2 = 0 ⇐ John Lindsay Orr Hilbert’s Tenth Problem

  70. Introduction Working with Diophantine Sets Sketch of Proof Coding n -tuples Going Into the Details Some Basic Diophantine Sets The set { ( a , b ) : aRb } is diophantine when “ R ” is one of the relations: a = b (consider “ ∃ x s.t. x + ( a − b ) 2 = 0”) a < b (consider “ ∃ x s.t. a + x + 1 = b ”) a | b (consider “ ∃ x s.t. ax = b ”) The set { ( a , b , c ) : a = rem ( b , c ) } is diophantine. Proof. a = rem ( b , c ) ⇐ ⇒ a < c & c | b − a ⇒∃ x , y s.t. ( a + x + 1 − b ) 2 + ( cy − ( b − a )) 2 = 0 ⇐ John Lindsay Orr Hilbert’s Tenth Problem

  71. Introduction Working with Diophantine Sets Sketch of Proof Coding n -tuples Going Into the Details The set { ( a , b , c ) : a ≡ b ( mod c ) } is diophantine. Proof. a ≡ b ( mod c ) ⇐ ⇒ rem ( a , c ) = rem ( b , c ) ⇐ ⇒∃ v , w s.t. v = rem ( a , c ) & w = rem ( b , c ) & w = v ⇒∃ v , w , x , y , x ′ , y ′ , z s.t. (( v + x + 1 − a ) 2 + ( cy − ( a − v )) 2 ) 2 ⇐ + (( w + x ′ + 1 − b ) 2 + ( cy ′ − ( b − w )) 2 ) 2 + ( z + ( v − w ) 2 ) 2 = 0 John Lindsay Orr Hilbert’s Tenth Problem

  72. Introduction Working with Diophantine Sets Sketch of Proof Coding n -tuples Going Into the Details The set { ( a , b , c ) : a ≡ b ( mod c ) } is diophantine. Proof. a ≡ b ( mod c ) ⇐ ⇒ rem ( a , c ) = rem ( b , c ) ⇐ ⇒∃ v , w s.t. v = rem ( a , c ) & w = rem ( b , c ) & w = v ⇒∃ v , w , x , y , x ′ , y ′ , z s.t. (( v + x + 1 − a ) 2 + ( cy − ( a − v )) 2 ) 2 ⇐ + (( w + x ′ + 1 − b ) 2 + ( cy ′ − ( b − w )) 2 ) 2 + ( z + ( v − w ) 2 ) 2 = 0 John Lindsay Orr Hilbert’s Tenth Problem

  73. Introduction Working with Diophantine Sets Sketch of Proof Coding n -tuples Going Into the Details The set { ( a , b , c ) : a ≡ b ( mod c ) } is diophantine. Proof. a ≡ b ( mod c ) ⇐ ⇒ rem ( a , c ) = rem ( b , c ) ⇐ ⇒∃ v , w s.t. v = rem ( a , c ) & w = rem ( b , c ) & w = v ⇒∃ v , w , x , y , x ′ , y ′ , z s.t. (( v + x + 1 − a ) 2 + ( cy − ( a − v )) 2 ) 2 ⇐ + (( w + x ′ + 1 − b ) 2 + ( cy ′ − ( b − w )) 2 ) 2 + ( z + ( v − w ) 2 ) 2 = 0 John Lindsay Orr Hilbert’s Tenth Problem

  74. Introduction Working with Diophantine Sets Sketch of Proof Coding n -tuples Going Into the Details Exponentiation is Diophantine Theorem (Matiyasevich, 1970) The set { ( a , b , c ) : a = b c } is diophantine. Corollary The set { ( a , n ) : a = n ! } is diophantine. a is prime ⇐ ⇒ a > 1 & gcd ( a , ( a − 1 )!) = 1 John Lindsay Orr Hilbert’s Tenth Problem

  75. Introduction Working with Diophantine Sets Sketch of Proof Coding n -tuples Going Into the Details Exponentiation is Diophantine Theorem (Matiyasevich, 1970) The set { ( a , b , c ) : a = b c } is diophantine. Corollary The set { ( a , n ) : a = n ! } is diophantine. a is prime ⇐ ⇒ a > 1 & gcd ( a , ( a − 1 )!) = 1 John Lindsay Orr Hilbert’s Tenth Problem

  76. Introduction Working with Diophantine Sets Sketch of Proof Coding n -tuples Going Into the Details Exponentiation is Diophantine Theorem (Matiyasevich, 1970) The set { ( a , b , c ) : a = b c } is diophantine. Corollary The set { ( a , n ) : a = n ! } is diophantine. a is prime ⇐ ⇒ a > 1 & gcd ( a , ( a − 1 )!) = 1 John Lindsay Orr Hilbert’s Tenth Problem

  77. Introduction Working with Diophantine Sets Sketch of Proof Coding n -tuples Going Into the Details Outline Introduction 1 Disclaimer History and Statement of the Problem Sketch of Proof 2 Turing Machines and Decidability Diophantine Sets Universal Diophantine Equations Going Into the Details 3 Working with Diophantine Sets Coding n -tuples John Lindsay Orr Hilbert’s Tenth Problem

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