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In early September, two mathematicians with the power of computers, announced that they finally solved the puzzle mathematicians for 65-year-old and 42 cubic mystery (see: "the mystery 42 has finally been cracked!"). The next time they say they are most interested in is non-trivial solution of the number 3, but less than a month, they will find the answer you want. In earlier times, the other two mathematicians proved the rational numbers associated with a guess (see: "troubled mathematician nearly 80 years of irrational numbers problem has been certified"). We are pleased to see these mathematical progress, but at the same time can not help but think of some mathematical problems have existed for hundreds of years, it is still challenging the wisdom of mankind. Some questions seem simple, but to prove that they uphill task. Below, we'll take a look at a few of these mathematical puzzles.
1. π + = and?
π and e are two constants in mathematics best known, but when they add up, but it became a problem everyone stumped.
The puzzles and the number of real algebraic related. If a real number is an integer coefficients of the polynomial roots, then we can say that the real number is algebraic . E.g. x²-6 is a polynomial with integer coefficients, since 1 and -6 are integers. x²-6 = 0 is a root of x = ± √6, which means are -√6 √6 and algebraic. All rational numbers, and the root of rational numbers are algebraic numbers. So you may feel that "most" real numbers are algebraic numbers. However, the result was just the opposite, "algebraic" is the opposite of " transcendental numbers ", the fact that almost all real numbers are transcendental number. Here, "almost all" is the mathematical meaning, then what is the number of algebra, which is transcendental?
π is a real number has been around for a long time, e is about to become known in the 17th century. For such two familiar numbers, you might think we know any of the fundamental problems associated with them.
The fact is, we know that π and e are transcendental numbers, but do not know π + e is algebraic or transcendental number. Similarly, we do not know the simple combination between πe, π / e and the other two numbers are what count. So in mathematics, there is a so some of us already know the numbers for hundreds of years or even thousands of years, contains some basic questions elusive.
2. γ is rational it? This is another easy but very difficult to write up the solution of the problem. Everything you need to know is rational definition.
It can be written as a rational number p / q in the form of numbers, where p and q are integers. Therefore 42,11 / 3, are rational; [pi] and √2 is irrational. This is a very basic nature, so you might think that we can easily determine whether a number is rational.
But let us come to know about Euler's constant - gamma . This is a real number, is approximately equal to 0.5772, the figure is represented by the equation γ closed type.
Use words to express that: "γ is the harmonic series and the natural logarithm of difference limit." So it is a combination of two already well-understood mathematical object, it can use other simple closed-form expression He appeared in hundreds of formulas.
But I do not know why, but why we do not know whether γ is rational. We have reached its hundreds of billions of digits, but still can not prove that it is rational. One theory is that, γ is an irrational number. And on a problem in π + e is like, this is another we can not answer the basic properties about a familiar figure.
3. anastomosis logarithm problem
Source: JJ Harrison / Wikimedia Commons
A broad class of problems in mathematics, called the sphere filled with problems. Whether all of these problems in pure mathematics or practical applications. In mathematics, the issues it deals with the accumulation of a sphere within a given space, and a real-life example is the grocery store fruit piled high. Some of these problems have a complete solution, but a few simple questions made us confused, such as anastomosis number of issues .
When a bunch of balls gathered in an area, then kiss each ball has a number that represents the number of the contact with the ball the other balls. If adjacent to a ball ball for six, then it anastomosis number is 6. There will be a bunch of balls kiss average number of connections, this number will help us to mathematically describe this situation. However, a fundamental problem with the number of connections relating to kiss, still unanswered.
First of all, we need to dimensions make some explanation. Dimension having a specific meaning in mathematics: they are independent axes. x-axis and y-axis represents the two dimensions of the coordinate plane. So whenever science fiction movie character says when they go to a different dimension, these words are mathematically meaningless, because you can not "go to the x-axis."
We know that one dimension is a straight line, is a two-dimensional plane. These lower values for the dimensions, the mathematicians have proven that the maximum number of anastomosis of these dimensions sphere may be present. It is a linear dimension on each of a left and right sides 2--. The exact number of connections kiss three-dimensional space until the 1950s before they get proof.
Kiss the number of connected issues beyond the three-dimensional hardly been resolved. Mathematicians are now slowly reduced to the possibility of a fairly narrow range - up to 24 dimensions, where the number of connections to kiss some dimensions are known. For larger or the number of general form, the openness of this issue is still very great. The presence of several major obstacle on the road to obtain a complete solution, including restrictions on computing power. Therefore, expected in the next few years, this problem will be able to gradually make progress. 4. Solutions of the problem junction
Source: Wikimedia Commons
End solution question the simplest versions has been resolved, but can not be fully resolved.
The problem with knot theory related to the idea of it is to try to use the formal mathematical methods (such as proof) to knot (such as shoelaces).
For example, you may know how to play a "square kink" and "outer parallel knot." Knotting step thereof as long as one of the twisted square knot in opposite directions parallel to the outer play a junction can be obtained. But can you prove that these results are different from you? Knot theory they can.
Square kink (a) parallel to the junction with the outer (lower). Source: Wikimedia Commons
A major problem kink theorists to deal with is the study of an algorithm to determine whether some confusion entanglement is a real kink, or it can be lifted entangled. The good news is that mathematicians have successfully written out of this algorithm in the past 20 years.
End solution of the problem is still calculated. It is an NP (nondeterministic polynomial) kind of problem, but we do not know if it is a P class of problems. This means that the current situation is that we have known these algorithms can handle any complexity of disentangling problem, but when they become more and more complex, time to deal with this issue on the president to incredible.
If someone can put forward an algorithm that can untie any knot in the so-called polynomial time, then untie the knot problem can be completely resolved. In addition, if someone can prove that this is not possible, then computationally intensive problem vast which means disentangling the problems faced it is inevitable.
The large base issue
Source: Wikimedia Commons
The late 19th century, the German mathematician Georg Cantor (Georg Cantor) found infinity of different sizes are present, he proved that an infinite collection of some of the elements contained more than other infinite sets.
Smallest infinite set can be represented ℵ₀, which is the size of the set of natural numbers, can be written ℕ = ℵ₀. Next it is ℵ₀ more common than infinite sets, such as Cantor proved the set of real numbers greater than ℵ₀ that ℝ> ℵ₀. But the set of real numbers is not so much, this is just the beginning of infinity.
Mathematicians are continuing to find more and more of infinity, or we can call it a large base . This is a purely mathematical process, if someone says, "I think the definition of the base, I can attest to this base larger than all the known base," Well, if he proves to be correct, then this would be known the maximum cardinality. Until someone comes up with a bigger.
Throughout the 20th century, a large base in the territory continue to move forward, and now the Wikipedia even has a "base" entries, there are many famous base are based on Cantor's named after. Well, all this will end it? The answer is almost certainly, although it will become very complicated.
In a sense, a huge base of top grade at hand. Some have been proven theorem of possible base provides some upper limit. But there are still many unanswered questions, some of the latest base until 2019 was finalized. The next few decades, we are likely to find more base. I hope we can get a final list covered a large base.
6. Goldbach's Conjecture
Source: Wikimedia Commons
Among the many mysteries of mathematics, some of the most difficult issues may also be able to describe in simple text, such as the Goldbach conjecture , it says that: "Every even number greater than 2 are two prime numbers and . "you can quickly check the lower number in mind with this: 13 + 18 = 23 + 19 = 5,42. Computer verification of this conjecture has been extended to a very large magnitude, but even so, we can show that the lack of proof for all natural numbers are established.
Goldbach's conjecture from the German mathematician Christian 1742 Sebastian Goldbach (Christian Goldbach) and legendary Swiss mathematician Leonhard Euler correspondence between (Leonhard Euler), Euler said: " I think (it) is a fully established theorem, even though I can not prove it. "Euler probably already aware of what makes this problem so difficult to solve. For larger numbers, which is written and the more the sum of two primes manner. 8 as 3 and 5 can be split into two prime numbers, and, it can be decomposed into 5 42 + 37,11 + 31,13 + 29,19 + 23. So for those very large numbers, the Goldbach conjecture is still not a full statement. Until now, mathematicians are still unable to fully prove certificate Goldbach conjecture, it is one of the oldest of all the mathematics open-ended questions.
Reference Links: https://www.popularmechanics.com/science/math/g29251596/impossible-math-problems/