Let be an integer. An -tuple of not necessarily different positive integers is expensive if there exists a positive integer such that a) Find all integers for which there exists an expensive -tuple.

b) Prove that for every odd positive integer there exists an integer such that belongs to an expensive -tuple.

There are exactly factors in the product on the left hand side.

On the graphic of the function were selected pairwise distinct points, abscissas of which are integer numbers from the segment . Prove that it is possible to choose six different selected points , , , , , such that areas of triangles and are equals.
A. Tereshin

Let be a graph colored using colors. We say that a vertex is forced if it has neighbors in all the other colors.

Prove that for any -regular graph that contains no triangles or quadrilaterals, there exists a coloring using colors such that at least of the colors have a forced vertex of that color.

Note: The graph coloring must be valid, this means no vertices of the same color may be adjacent.

For each integer sequence , a single parity swapping is to choose terms in this sequence, say and , such that is odd, then switch their placement, while the other terms stay in place. This creates a new sequence.

Find the minimal number of single parity swapping to transform the sequence to , using only single parity swapping.

The P vs. NP Problem: One of the Greatest Unsolved Questions in Computer Science

The P vs. NP problem is one of the most profound and long-standing unsolved problems in mathematics and theoretical computer science. It is one of the seven Millennium Prize Problems, meaning that a correct proof (or disproof) earns a reward of $1,000,000 from the Clay Mathematics Institute.

At its core, the P vs. NP problem asks:

Is every problem whose solution can be verified quickly also solvable quickly?

More formally:

Does P = NP?

If the answer is "yes," it means that problems for which a solution can be verified quickly (in polynomial time) can also be solved quickly. If "no," then there are problems that are inherently hard to solve, even though checking a solution is easy.


[center]IMAGE[/center]


[center]Euler diagram for P, NP, NP-complete, and NP-hard set of problems (excluding the empty language and its complement, which belong to P but are not NP-complete)[/center]

1. Understanding P and NP

In complexity theory, problems are classified based on how efficiently they can be solved by an algorithm. The classes P and NP describe two fundamental categories of decision problems.

[list]
[*] P (Polynomial Time): This is the class of decision problems that can be solved by a deterministic Turing machine in polynomial time. In other words, if a problem is in P, there exists an algorithm that can solve it in time bounded by a polynomial function of the input size.

Examples of problems in P include:
[list]
[*] Sorting a list (using algorithms like merge sort).
[*] Finding the greatest common divisor (using the Euclidean algorithm).
[*] Determining whether a number is prime (with modern algorithms like AKS primality testing).
[/list]

[*] NP (Nondeterministic Polynomial Time): This is the class of decision problems where a proposed solution can be verified in polynomial time by a deterministic Turing machine. An equivalent definition is that NP problems can be solved by a nondeterministic Turing machine in polynomial time.

Examples of problems in NP include:
[list]
[*] The Traveling Salesman Problem (TSP): Given a list of cities and distances between them, is there a tour visiting each city exactly once with a total length less than a given value?
[*] The Boolean Satisfiability Problem (SAT): Given a Boolean formula, is there an assignment of variables that makes the formula true?
[*] Graph Coloring: Can the vertices of a graph be colored with colors such that no two adjacent vertices share the same color?
[/list]
[/list]

By definition, we have:


The open question is whether this inclusion is strict: Is P = NP, or is P NP?

2. NP-Complete Problems: The Hardest Problems in NP

A subset of NP problems, called NP-complete problems, are the "hardest" problems in NP. If any NP-complete problem can be solved in polynomial time, then P = NP.

To formally define NP-complete problems:

A problem is NP-complete if:

[list]
[*] (it is in NP, meaning solutions can be verified in polynomial time).
[*] Every other problem in NP can be reduced to in polynomial time (this means if you can solve efficiently, you can solve all NP problems efficiently).
[/list]

The first NP-complete problem, Boolean satisfiability (SAT), was proved by Stephen Cook in 1971 through the famous Cook-Levin theorem. Since then, thousands of problems have been shown to be NP-complete.

Examples of NP-complete problems:

[list]
[*] SAT (Boolean Satisfiability Problem).
[*] Traveling Salesman Problem (decision version).
[*] 3-Colorability (can a graph be colored with 3 colors?).
[*] Subset Sum Problem (is there a subset of numbers that sums to a target value?).
[/list]

3. Implications of P = NP or P ≠ NP

The resolution of the P vs. NP problem would have enormous implications across mathematics, computer science, cryptography, and more.

If P = NP:

[list]
[*] Every problem for which a solution can be verified quickly can also be solved quickly.
[*] Many currently hard problems (such as breaking cryptographic codes) would become easy.
[*] Modern encryption methods based on the hardness of NP problems (like RSA) would become insecure.
[*] Solutions to many practical optimization problems would become feasible in real time.
[/list]

If P ≠ NP:

[list]
[*] There exist problems in NP that are inherently hard to solve, even though their solutions can be verified efficiently.
[*] Cryptographic systems would remain secure.
[*] Certain problems (such as protein folding, perfect route planning) will likely remain computationally infeasible to solve exactly.
[/list]

4. Attempts to Solve the P vs. NP Problem

Despite extensive efforts, no one has been able to prove or disprove whether P = NP. Some major developments include:

[list]
[*] Cook-Levin Theorem (1971): Stephen Cook and independently Leonid Levin proved that SAT is NP-complete, introducing the entire field of NP-completeness.
[*] Karp’s 21 Problems (1972): Richard Karp showed that 21 classical problems (including TSP and graph coloring) are NP-complete.
[*] Cryptographic Evidence: Many encryption systems rely on the assumption that P ≠ NP, though this is not a proof.
[*] Relativization (Baker, Gill, and Solovay – 1975): Certain techniques (oracle machines) cannot resolve P vs. NP, suggesting new methods are needed.
[/list]

5. Theoretical and Practical Consequences

If P = NP, it would revolutionize fields such as:

[list]
[*] Cryptography: Encryption systems would collapse, making secure communication impossible.
[*] Artificial Intelligence: Efficient solutions to complex problems like natural language understanding and protein folding would become possible.
[*] Optimization: Problems like airline scheduling and supply chain management would become trivial to solve.
[/list]

If P ≠ NP, it would confirm the inherent hardness of many problems and validate the foundation of computational security.

6. Summary

[list]
[*] P vs. NP asks whether every problem whose solution can be verified in polynomial time can also be solved in polynomial time.
[*] If P = NP, many hard problems would become easy to solve, impacting encryption and optimization.
[*] If P ≠ NP, some problems remain inherently difficult to solve efficiently.
[*] The P vs. NP problem remains unsolved and is one of the most important open questions in computer science and mathematics.
[/list]

7. References


[center][youtube]https://www.youtube.com/watch?v=pQsdygaYcE4[/youtube][/center]

[list]
[*] Clay Mathematics Institute: P vs NP Problem
[*] Arora, S., & Barak, B. Computational Complexity: A Modern Approach.
[*] Garey, M. R., & Johnson, D. S. Computers and Intractability: A Guide to the Theory of NP-Completeness.
[*] Wikipedia: P vs NP Problem
[*] AoPS: P vs NP Problem
[/list]

Exploring Fascinating Math Conjectures: A Journey into the Unknown

Mathematics is a field that is full of beautiful puzzles, some of which have remained unsolved for centuries. These unsolved problems, or conjectures, challenge mathematicians to delve deeper into the abstract world of numbers, shapes, and logic. In this blog, we'll explore five of the most intriguing mathematical conjectures, and break down their significance in the world of mathematics.

1. The Riemann Hypothesis

The Riemann Hypothesis, proposed by Bernhard Riemann in 1859, is one of the most famous and long-standing conjectures in mathematics. It is a statement about the distribution of prime numbers, which is one of the central topics in number theory. To understand the conjecture, we first need to introduce the concept of the Riemann zeta function:

for complex numbers with a real part greater than 1. This infinite series converges for , but the Riemann zeta function is also analytically continued to other values of , except for , where it has a pole.

The Riemann Hypothesis posits that all **non-trivial zeros** of the Riemann zeta function, the values of where , lie on the "critical line," where the real part of is . In other words, all such zeros should be of the form , where is a real number.

The conjecture is deeply important because the distribution of these zeros is intimately tied to the distribution of prime numbers. The location of the zeros gives us insights into how prime numbers are spread along the number line. The truth of the Riemann Hypothesis would lead to breakthroughs in prime number theory, cryptography, and many other areas of mathematics.

2. The Collatz Conjecture (3x+1 Problem)

The Collatz Conjecture, sometimes called the "3x+1 problem," is an elementary-looking problem that has stumped mathematicians for decades. It starts with any positive integer and applies the following steps:
[list]
[*]If is even, divide it by 2.
[*]If is odd, multiply it by 3 and add 1.
[/list]

Repeat the process with the resulting number. The conjecture asserts that no matter what positive integer you start with, the sequence will always eventually reach 1.

For example, starting with :
While this looks simple, the Collatz Conjecture remains unsolved. The sequence has been verified for a large range of numbers, but a general proof that every positive integer eventually reaches 1 is still elusive. No one has been able to prove that the sequence always terminates, nor has anyone been able to find a counterexample. The conjecture is puzzling because it involves very basic operations but exhibits unpredictable behavior.

Mathematically, the conjecture touches on dynamics, number theory, and iterated functions, but its ultimate resolution remains one of the great mysteries of mathematics.

3. The Goldbach Conjecture

The Goldbach Conjecture, proposed by Christian Goldbach in 1742, is one of the oldest and most famous unsolved problems in number theory. It posits that:


For example:
[list]
[*]4 = 2 + 2
[*]6 = 3 + 3
[*]8 = 3 + 5
[*]10 = 5 + 5
[/list]

The conjecture has been tested for very large numbers, and no counterexample has been found. In fact, it is believed that every even number greater than 2 can be written as the sum of two primes, but a formal proof is still missing.

The conjecture has important implications for understanding the additive structure of prime numbers. It suggests that primes are much more prevalent in our number system than might first be expected. Mathematicians have attempted many approaches to prove Goldbach’s Conjecture, including using powerful tools from analytic number theory, but a solution remains elusive.

4. The P vs NP Problem

The P vs NP Problem is one of the seven Millennium Prize Problems and has profound implications for computer science and mathematics. It asks whether the class of problems that can be solved efficiently (in polynomial time) is the same as the class of problems whose solutions can be verified efficiently.

Let’s define the terms:
[list]
[*]P represents the class of problems that can be solved in polynomial time (i.e., there is an algorithm that can find the solution in time proportional to a polynomial function of the input size).
[*]NP represents the class of problems for which a proposed solution can be verified in polynomial time (i.e., if someone gives you a potential solution, you can check it in polynomial time).
[/list]

The P vs NP problem asks whether every problem for which a solution can be verified quickly (i.e., in polynomial time) can also be solved quickly. In other words, is P equal to NP?

If P = NP, it would imply that many problems we currently think are difficult to solve could actually be solved quickly, revolutionizing fields like cryptography, optimization, and artificial intelligence. On the other hand, if P ≠ NP, it would affirm that there are problems that, while easy to check, are inherently difficult to solve. Despite significant effort, no one has yet been able to prove whether P = NP or P ≠ NP, making this one of the most profound open questions in mathematics.

5. The Twin Prime Conjecture

The Twin Prime Conjecture posits that there are infinitely many pairs of prime numbers that differ by exactly 2. These pairs are known as twin primes. Some examples include:
[list]
[*](3, 5)
[*](5, 7)
[*](11, 13)
[*](17, 19)
[/list]

The conjecture was first proposed by the mathematician Alphonse de Polignac in 1846. It suggests that for every large number , there will always be some twin prime pair larger than .

While the conjecture has not been proven, a number of important results have been obtained in its study. In 2013, mathematician Yitang Zhang made a breakthrough by showing that there are infinitely many pairs of primes that differ by at most 70 million. While this does not directly prove the Twin Prime Conjecture, it was the first time anyone had shown that there is a bounded gap between prime numbers. Since then, other mathematicians have continued to refine this bound.

The conjecture is closely related to the distribution of prime numbers and has been a subject of intense study for over a century.

Conclusion: The Fascinating Nature of Conjectures

Mathematical conjectures like these represent the cutting edge of mathematical discovery. They challenge our understanding of numbers, shapes, and functions, and they inspire mathematicians to dig deeper into the very foundations of mathematics. Some of these conjectures have resisted proof for centuries, while others have seen recent breakthroughs, yet all of them remain crucial to the advancement of mathematical theory. The pursuit of their resolution continues to drive progress in both pure mathematics and practical applications.

Who knows? The next breakthrough might be just around the corner, waiting to be discovered.

Feel free to share your thoughts and any other conjectures that interest you in the comments below!