SOME FAMOUS UNSOLVED PROBLEMS. November 18, / 5

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1 SOME FAMOUS UNSOLVED PROBLEMS November 18, / 5

2 SOME FAMOUS UNSOLVED PROBLEMS Goldbach conjecture: Every even number greater than four is the sum of two primes. November 18, / 5

3 SOME FAMOUS UNSOLVED PROBLEMS Goldbach conjecture: Every even number greater than four is the sum of two primes. Goldbach, 1742, in letter to Euler November 18, / 5

4 SOME FAMOUS UNSOLVED PROBLEMS Goldbach conjecture: Every even number greater than four is the sum of two primes. Goldbach, 1742, in letter to Euler Schnirelman, 1939: every even number can be written as a sum of not more than 300,000 primes November 18, / 5

5 SOME FAMOUS UNSOLVED PROBLEMS Goldbach conjecture: Every even number greater than four is the sum of two primes. Goldbach, 1742, in letter to Euler Schnirelman, 1939: every even number can be written as a sum of not more than 300,000 primes Known to be true for numbers November 18, / 5

6 Twin Prime conjecture: There are infinitely many pairs (n, n + 2) where both n and n + 2 are prime. November 18, / 5

7 Twin Prime conjecture: There are infinitely many pairs (n, n + 2) where both n and n + 2 are prime. Conjectured by Euclid November 18, / 5

8 Twin Prime conjecture: There are infinitely many pairs (n, n + 2) where both n and n + 2 are prime. Conjectured by Euclid One recent example: ( ) ± 1 November 18, / 5

9 Twin Prime conjecture: There are infinitely many pairs (n, n + 2) where both n and n + 2 are prime. Conjectured by Euclid One recent example: ( ) ± 1 Extended conjecture (de Polignac, 1849) : For each positive integer k there are infinitely many pairs of primes [p, q] such that p q = 2k. November 18, / 5

10 Twin Prime conjecture: There are infinitely many pairs (n, n + 2) where both n and n + 2 are prime. Conjectured by Euclid One recent example: ( ) ± 1 Extended conjecture (de Polignac, 1849) : For each positive integer k there are infinitely many pairs of primes [p, q] such that p q = 2k. In 2013 Yitang Zhang, (Ph. D. Purdue, 1991, employee of Subway in the 90 s, instructor at the Univeristy of New Hampshire since 1999, proved that this is true for some value of k. His estimate was: k 70, 000, 000. It is now known that k < 245. k = 2 is the twin prime conjecture. November 18, / 5

11 Twin Prime conjecture: There are infinitely many pairs (n, n + 2) where both n and n + 2 are prime. Conjectured by Euclid One recent example: ( ) ± 1 Extended conjecture (de Polignac, 1849) : For each positive integer k there are infinitely many pairs of primes [p, q] such that p q = 2k. In 2013 Yitang Zhang, (Ph. D. Purdue, 1991, employee of Subway in the 90 s, instructor at the Univeristy of New Hampshire since 1999, proved that this is true for some value of k. His estimate was: k 70, 000, 000. It is now known that k < 245. k = 2 is the twin prime conjecture. Zhang was awarded the most important American math prizes in his area, and a MacArthur Fellowship. He became a full professor at UNH in January, November 18, / 5

12 Collatz Conjecture (1937). November 18, / 5

13 Clay Millenium problems $1,000,000 prize for each solution. November 18, / 5

14 Clay Millenium problems $1,000,000 prize for each solution. Riemann hypothesis: The Riemann zeta function ζ (z) = n=1 1 n z has all its zeros on the line x = 2 1, where z = x + iy. (Important implications in number theory) November 18, / 5

15 Clay Millenium problems $1,000,000 prize for each solution. Riemann hypothesis: The Riemann zeta function ζ (z) = n=1 1 n z has all its zeros on the line x = 2 1, where z = x + iy. (Important implications in number theory) P = NP (A computer science problem) November 18, / 5

16 Clay Millenium problems $1,000,000 prize for each solution. Riemann hypothesis: The Riemann zeta function ζ (z) = n=1 1 n z has all its zeros on the line x = 2 1, where z = x + iy. (Important implications in number theory) P = NP (A computer science problem) Problems in NP have the property that their answers can be checked in a relatively short time ( polynomial time ). Problems in P have the property that answers can be found in a relatively short time. November 18, / 5

17 Clay Millenium problems $1,000,000 prize for each solution. Riemann hypothesis: The Riemann zeta function ζ (z) = n=1 1 n z has all its zeros on the line x = 2 1, where z = x + iy. (Important implications in number theory) P = NP (A computer science problem) Problems in NP have the property that their answers can be checked in a relatively short time ( polynomial time ). Problems in P have the property that answers can be found in a relatively short time. Example: Factoring numbers is thought to be in NP but not in P. November 18, / 5

18 Clay Millenium problems $1,000,000 prize for each solution. Riemann hypothesis: The Riemann zeta function ζ (z) = n=1 1 n z has all its zeros on the line x = 2 1, where z = x + iy. (Important implications in number theory) P = NP (A computer science problem) Problems in NP have the property that their answers can be checked in a relatively short time ( polynomial time ). Problems in P have the property that answers can be found in a relatively short time. Example: Factoring numbers is thought to be in NP but not in P. Yang-Mills and mass gap November 18, / 5

19 Clay Millenium problems $1,000,000 prize for each solution. Riemann hypothesis: The Riemann zeta function ζ (z) = n=1 1 n z has all its zeros on the line x = 2 1, where z = x + iy. (Important implications in number theory) P = NP (A computer science problem) Problems in NP have the property that their answers can be checked in a relatively short time ( polynomial time ). Problems in P have the property that answers can be found in a relatively short time. Example: Factoring numbers is thought to be in NP but not in P. Yang-Mills and mass gap regularity for the Navier-Stokes equations of fluid motion in three dimensions November 18, / 5

20 Clay Millenium problems $1,000,000 prize for each solution. Riemann hypothesis: The Riemann zeta function ζ (z) = n=1 1 n z has all its zeros on the line x = 2 1, where z = x + iy. (Important implications in number theory) P = NP (A computer science problem) Problems in NP have the property that their answers can be checked in a relatively short time ( polynomial time ). Problems in P have the property that answers can be found in a relatively short time. Example: Factoring numbers is thought to be in NP but not in P. Yang-Mills and mass gap regularity for the Navier-Stokes equations of fluid motion in three dimensions Hodge conjecture November 18, / 5

21 Clay Millenium problems $1,000,000 prize for each solution. Riemann hypothesis: The Riemann zeta function ζ (z) = n=1 1 n z has all its zeros on the line x = 2 1, where z = x + iy. (Important implications in number theory) P = NP (A computer science problem) Problems in NP have the property that their answers can be checked in a relatively short time ( polynomial time ). Problems in P have the property that answers can be found in a relatively short time. Example: Factoring numbers is thought to be in NP but not in P. Yang-Mills and mass gap regularity for the Navier-Stokes equations of fluid motion in three dimensions Hodge conjecture Birch and Swinnerton-Dyer conjecture November 18, / 5

22 Poincaré conjecture November 18, / 5

23 Poincaré conjecture Henri Poincaré, November 18, / 5

24 Poincaré conjecture Henri Poincaré, Proved by G. Perelman (1966- ) in 2007, November 18, / 5

25 Poincaré conjecture Henri Poincaré, Proved by G. Perelman (1966- ) in 2007, but he would not accept the award November 18, / 5