Unsolved Problems

Happy End Problem

Proving that the _____ ___ [g(n) points will always have no three points that are colinear and vertices of a convex polygon of n sides] that problem can be done using (n).

Cubic Numbers

Proving which numbers can be represented as the sum of three or four cubic numbers. It is known that every integer is a sum of at most 5 signed cube, so it is believed it can be reduced to four, then three.

Solved or Unsolved:The conjecture that there exists a Hadamard matrix for every positive multiple of 4.

Unsolved

Perfect numbers

Whether any odd perfect numbers exist. A perfect number is the sum of its devisors. All numbers from 10^300 have been checked.

Lehmer's Totient Problem

_____ ___ problem asks if there exist any composite numbers n such that phi(n)|(n-1), where phi(n) is the totient function? n would have to be a Carmichael number.

Millenium Prize

7 problems stated by Clay Mathematics Institute in 2000. One, the Poincare conjecture, has been solved.

Are NP problems P problems?

A P problem (solution set bound by a polynomial time) is also always a NP problem (nondeterministic polynomial time). Linear programming was found to be P. If a problem is NP, but P, it takes an exhaustive search to find the answer. Ergo, this is an important unsolved problem.

Lehmer's Mahler Measure Problem

An unsolved problem in mathematics attributed to Lehmer (1933) that concerns the minimum _____ ___ M_1(P) for a univariate polynomial P(x) that is not a product of cyclotomic polynomials.

Goldbach's Conjecture

Euler's re-expression: all positive even integers greater than or equal to 4 are the sum of two primes

Symmetric Group Probability

Finding a formula for the probability that two elements chosen at random generate the symmetric group S_n.

Collatz Problem

Let a_0 be an integer. Then one form of _____ problem asks if iterating a_n = {.5a_n-1 for a_n-1even; 3a_n-1 +1 for a_n-1 odd, it always returns to 1 for positive a_0.

196 Algorithm

Take any positive integer of two digits or more, reverse the digits, and add to the original number. This is the operation of the reverse-then-add sequence. Now repeat the procedure with the sum so obtained until a palindromic number is obtained. This procedure quickly produces palindromic numbers for most integers. 196 is the smallest number that does not behave like this, others being 295, 394, 493, 592, 689, 691, 788, 790, 879, 887.

Birch and Swinnerton-Dyer conjecture

The Birch and Swinnerton-Dyer conjecture deals with certain types of equations; The conjecture is that there is a simple way to tell whether such equations have a finite or infinite number of rational solutions of defining elliptic curves over the rational numbers.

Hodge Conjecture

The Hodge conjecture is that for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.

Navier-Stokes existence and smoothness

The Navier-Stokes equations describe the motion of fluids. Although they were first stated in the 19th century, they are still not well-understood. The problem is to make progress towards a mathematical theory that will give insight into these equations.

Twin Prime Conjecture

There are an infinite number of pairs of twin primes (p, p+2)

Euler Brick/Perfect Cuboid

cuboid with integer edges a>b>c and face diagonals: d_ab = radical a^2+b^2 d_ac = radical a^2+c^2 d_bc = radical b^2 + c^2 If the space diagonal is an integer, this is a perfect cuboid. This has never been found, ergo no ___ ___ have been found.

Goldbach's Original Conjecture

every number greater that 2 is the sum of 3 primes (1 is considered a prime here)

Euler-Mascheroni Constant

it is not known if y=.577215664 is irrational

Yang-Mills existence and mass gap

n physics, classical Yang-Mills theory is a generalization of the Maxwell theory of electromagnetism where the chromo-electromagnetic field itself carries charges. As a classical field theory it has solutions which travel at the speed of light so that its quantum version should describe massless particles (gluons). However, the postulated phenomenon of color confinement permits only bound states of gluons, forming massive particles. This is the mass gap. Another aspect of confinement is asymptotic freedom which makes it conceivable that quantum Yang-Mills theory exists without restriction to low energy scales. The problem is to establish rigorously the existence of the quantum Yang-Mills theory and a mass gap. The official statement of the problem was given by Arthur Jaffe and Edward Witten.[7]

Goldbach's Partition

p and q are primes, n is a positive integer: (p+q)=2n

Percolation Threshold

percolation threshold is used to denote the probability which "marks the arrival" (Grimmett 1999) of an infinite connected component (i.e., of a percolation) within a particular model. It is not known if it is possible to derive an analytic form for the square site percolation threshold.

Riemann Hypothesis

the Riemann zeta function nor any Dirichlet L-series has a zero with real part larger than 1/2.