http://kenji.blog/en/tags/unsolved-problems/Recent content in Unsolved Problems on kenji.blogHugo -- gohugo.ioenkenjinoteWed, 11 Sep 2024 02:22:39 +0900http://kenji.blog/en/p/pnp%E4%BA%88%E6%83%B3/Wed, 11 Sep 2024 02:22:39 +0900http://kenji.blog/en/p/pnp%E4%BA%88%E6%83%B3/<h1 id="overview">Overview </h1><div class="highlight"><div class="chroma"> <table class="lntable"><tr><td class="lntd"> <pre tabindex="0" class="chroma"><code><span class="lnt">1 </span></code></pre></td> <td class="lntd"> <pre tabindex="0" class="chroma"><code class="language-fallback" data-lang="fallback"><span class="line"><span class="cl">Class P is the class of problems that are decidable in polynomial time by a deterministic Turing machine. Class NP is the class of problems for which the correctness of a witness (evidence that the answer is Yes) can be verified in polynomial time when given the witness. Since problems decidable in polynomial time are also verifiable in polynomial time, it is obvious that P ⊆ NP. However, it is not clear whether P is a proper subset of NP. Although there is no proof yet, many researchers believe that P ≠ NP. This conjecture that class P and class NP are not equal is known as the &#34;P ≠ NP conjecture&#34;. </span></span></code></pre></td></tr></table> </div> </div><p>Reference site : <a class="link" href="https://daigakudenki.com/np-hard/" target="_blank" rel="noopener" >https://daigakudenki.com/np-hard/</a></p>http://kenji.blog/en/p/%E6%95%B0%E5%AD%A6%E3%81%AE%E6%9C%AA%E8%A7%A3%E6%B1%BA%E5%95%8F%E9%A1%8C/Sat, 02 Mar 2024 22:57:36 +0900http://kenji.blog/en/p/%E6%95%B0%E5%AD%A6%E3%81%AE%E6%9C%AA%E8%A7%A3%E6%B1%BA%E5%95%8F%E9%A1%8C/<img src="http://kenji.blog/p/%E6%95%B0%E5%AD%A6%E3%81%AE%E6%9C%AA%E8%A7%A3%E6%B1%BA%E5%95%8F%E9%A1%8C/img.png" alt="Featured image of post Unsolved Problems in Mathematics" /><h1 id="unsolved-problems-in-mathematics">Unsolved Problems in Mathematics </h1><p>This is an explanation of unsolved problems in mathematics. Although the problems themselves are simple, there are still many that have not been proven.</p> <h2 id="are-there-infinitely-many-perfect-numbers">Are there infinitely many perfect numbers? </h2><p>A perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For example,</p> <ul> <li>6 has divisors 1, 2, and 3, and 1+2+3=6, so it is a perfect number.</li> <li>28 has divisors 1, 2, 4, 7, and 14, and 1+2+4+7+14=28, so it is a perfect number.</li> </ul> <p>Currently, only 51 perfect numbers have been discovered. It is conjectured that there are infinitely many, but this has not yet been proven.</p> <h2 id="goldbachs-conjecture">Goldbach&rsquo;s Conjecture </h2><p>Goldbach&rsquo;s conjecture states that every even integer greater than 2 can be expressed as the sum of two primes. (Here, a prime number is a natural number that has no positive divisors other than 1 and itself.)</p> <p>For example,</p> <ul> <li>4=2+2</li> <li>6=3+3</li> <li>8=3+5</li> <li>10=3+7=5+5</li> </ul> <p>The problem itself is very simple, but it remains unproven.</p> <h2 id="riemann-hypothesis">Riemann Hypothesis </h2><p>The Riemann hypothesis is the conjecture that the zeros of the Riemann zeta function are restricted to negative even integers and complex numbers with real part 1 / 2.</p> <p>The Riemann zeta function is the function $\zeta$ defined by:</p> $$\zeta(s):=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=1+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+{\frac {1}{4^{s}}}+\cdots$$ <p>where $s$ is a complex number and $n$ is a natural number.</p>