Mathematical problems

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Solved problems

There are many unsolved problems in mathematics. Severalfamous problems which have recently been solved include: 1. The Pólya conjecture (disproven byHaselgrove 1958, smallest counterexample found by Tanaka 1980). 2. The four-color theorem (Appel and Haken1977ab and Appel et al. 1977 using a computer-assisted proof). 3. The Bieberbach conjecture (L. deBranges 1985). 4. Tait's flyping conjecture (Menasco and Thistlethwaite in 1991) and the other two of Tait's knot conjectures (by various authors 1987). 5. Fermat's last theorem (Wiles 1995, Taylorand Wiles 1995). 6. The Kepler conjecture (Hales 2002). 7. The Taniyama-Shimura conjecture(Breuil et al. in 1999). 8. The honeycomb conjecture (Hales 1999).9. The Poincaré conjecture. 10. Catalan's conjecture. 11. The strong perfect graph theorem...

Unsolved problems

There are many unsolved problems in mathematics. Some prominent outstanding unsolved problems (as well as some which are not necessarily so well known) include 1. The Goldbach conjecture. 2. The Riemann hypothesis. 3. The conjecture that there exists a Hadamard matrixfor every positive multiple of 4. 4. The twin prime conjecture (i.e., the conjecture that there are an infinite number of twin primes). 5. Determination of whether NP-problems are actuallyP-problems. 6. The Collatz problem. 7. Proof that the 196-algorithm does not terminatewhen applied to the number 196. 8. Proof that 10 is a solitary number. 9. Finding a formula for the probability that two elements chosen at random generate the symmetric group . 10. Solving the happy end problem for arbitrary . 11. Finding an Euler brick whose space diagonal isalso an integer. 12. Proving which numbers can be represented as a sum of three or four (positiveor negative) cubic numbers. 13. Lehmer's..

Smale's problems

Smale's problems are a list of 18 challenging problems for the twenty-first century proposed by Field medalist Steven Smale. These problems were inspired in part by Hilbert's famous list of problems presented in 1900 (Hilbert's problems), and in part in response to a suggestion by V. I. Arnold on behalf of the International Mathematical Union that mathematicians describe a number of outstanding problems for the 21st century.1. The Riemann hypothesis. 2. The Poincaré conjecture. 3. Does (i.e., are P-problems equivalent to NP-problems)? 4. Integer zeros of a polynomial. 5. Height bounds for Diophantine curves. 6. Finiteness of the number of relative equilibria in celestial mechanics. 7. Distribution of points on the 2-sphere. 8. Introduction of dynamics into economic theory. 9. The linear programming problem. 10. The closing lemma. 11. Is 1-dimensional dynamics generally hyperbolic? 12. Centralizers of diffeomorphisms...

Simon's problems

A set of 15 open problems on Schrödinger operators proposed by mathematical physicist Barry Simon (2000). This set of problems follows up a 1984 list of open problems in mathematical physics also proposed by Simon, of which thirteen involved Schrödinger operators.1. Extended states. Prove for and suitable values of that the Anderson model has purely absolutely continuous spectrum in some energy range. 2. Localization in two dimensions. Prove that for , the spectrum of the Anderson model is dense pure point for all values of . 3. Quantum diffusion. Prove that for and values of where there is a.c. spectrum that grows as as . 4. Ten Martini problem. Prove for all and all irrational that (which is independent) is a Cantor set, that is, that it is nowhere dense. 5. Prove for all irrational and that has measure zero. 6. Prove for all irrational and that the spectrum is purely absolutely continuous. 7. Do there exist potentials on so that..

Landau's problems

Landau's problems are the four "unattackable" problems mentioned by Landau in the 1912 Fifth Congress of Mathematicians in Cambridge, namely: 1. The Goldbach conjecture, 2. Twin prime conjecture, 3. Legendre's conjecture that for every there exists a prime between and (Hardy and Wright 1979, p. 415; Ribenboim 1996, pp. 397-398), and 4. The conjecture that there are infinitely many primes of the form (Euler 1760; Mirsky 1949; Hardy and Wright 1979, p. 19; Ribenboim 1996, pp. 206-208). The first few such primes are 2, 5, 17, 37, 101, 197, 257, 401, ... (OEIS A002496). Although it is not known if there always exists a prime between and , Chen (1975) has shown that a number which is either a prime or semiprime does always satisfy this inequality. Moreover, there is always a prime between and where (Iwaniec and Pintz 1984; Hardy and Wright 1979, p. 415). The smallest primes between and for , 2, ..., are 2, 5, 11,..

Hilbert's problems

Hilbert's problems are a set of (originally) unsolved problems in mathematics proposed by Hilbert. Of the 23 total appearing in the printed address, ten were actually presented at the Second International Congress in Paris on August 8, 1900. In particular, the problems presented by Hilbert were 1, 2, 6, 7, 8, 13, 16, 19, 21, and 22 (Derbyshire 2004, p. 377). Furthermore, the final list of 23 problems omitted one additional problem on proof theory (Thiele 2001).Hilbert's problems were designed to serve as examples for the kinds of problems whose solutions would lead to the furthering of disciplines in mathematics. As such, some were areas for investigation and therefore not strictly "problems."1. "Cantor's problem of the cardinal number of the continuum." The question of if there is a transfinite number between that of a denumerable set and the numbers of the continuum was answered by Gödel and Cohen in their..

Coin problem

Let there be integers with . The values represent the denominations of different coins, where these denominations have greatest common divisor of 1. The sums of money that can be represented using the given coins are then given by(1)where the are nonnegative integers giving the numbers of each coin used. If , it is obviously possibly to represent any quantity of money . However, in the general case, only some quantities can be produced. For example, if the allowed coins are , it is impossible to represent and 3, although all other quantities can be represented.Determining the function giving the greatest for which there is no solution is called the coin problem, or sometimes the money-changing problem. The largest such for a given problem is called the Frobenius number .The result(2)(3)(Nijenhuis and Wilf 1972) is mathematical folklore. The total number of such nonrepresentable amounts is given by(4)The largest nonrepresentable amounts for..

Lam's problem

Given a (0,1)-matrix, fill 11 spaces in each row in such a way that all columns also have 11 spaces filled. Furthermore, each pair of rows must have exactly one filled space in the same column. This problem is equivalent to finding a projective plane of order 10. Using a computer program, Lam et al. (1989) showed that no such arrangement exists.Lam's problem is equivalent to finding nine orthogonal Latinsquares of order 10.

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