Undecidability in number theory Listable sets DPRM theorem - PowerPoint PPT Presentation

Undecidability in number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Undecidability in number theory Listable sets DPRM theorem Consequences of DPRM Prime-producing Bjorn Poonen polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Hedrick Lecture 1 Status of knowledge MAA MathFest 2014 August 7, 2014

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 29? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 29? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems Yes: ( x , y , z ) = (3 , 1 , 1). H10 over Q First-order sentences Subrings of Q Status of knowledge

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 30? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 30? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Yes: ( x , y , z ) = ( − 283059965 , − 2218888517 , 2220422932). Subrings of Q Status of knowledge (discovered in 1999 by E. Pine, K. Yarbrough, W. Tarrant, and M. Beck, following an approach suggested by N. Elkies.)

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 33? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 33? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge Unknown.

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 47? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 51? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 84? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 54? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 11? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = − 98? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 427? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 689? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 138? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 823? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 549? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = − 190? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 3837? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 3992? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 5566? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge

Undecidability in Examples of polynomial equations number theory Bjorn Poonen H10 Polynomial equations Hilbert’s 10th problem Diophantine sets Do there exist integers x , y , z such that Listable sets DPRM theorem x 3 + y 3 + z 3 = 5172? Consequences of DPRM Prime-producing polynomials Riemann hypothesis Related problems H10 over Q First-order sentences Subrings of Q Status of knowledge