https://www.2021hermes.com/
People always talk about all kinds of unsolved mysteries. From Amelia Earhart's strange disappearance over the Pacific in 1937, to three prisoners, Frank Morris, John Anglin and Clarence in 1962 ·Clarence Anglin legendarily escaped from Alcatraz Island in California, USA. Various mysterious stories have enriched the boring lives of the public.
Of course, these stories(2021 hermes) are not only from real history. In 1979, Douglas Adams published the first of his five series of science fiction novels-"The Hitchhiker's Guide to the Galaxy". At the end of this novel, a supercomputer named "Thinking" reveals the answer to the "ultimate question" about "life, the universe, and everything": "42". It took 7.5 million years for "Thinking" to calculate this result. But the alien who created this supercomputer in the novel is disappointing. After all, a single number is not very useful. However, "Thinking" also told the aliens that the questions they raised were too general. To find an accurate statement of the problem, the supercomputer needs to spend a long time to update itself. The new version of the computer is the earth. Interested readers can read Adams' book. The number "42" then became the foundation of geek culture (counter-mainstream culture), which led to many allusions and jokes. For example, if you enter "What is the answer to everything?" in a search engine, most of the answers that pop up are "42". Use other languages (such as French or German) or different search engines to get the same results. Since 2013, a series of computer training schools named "42 Network" have been successively established all over the world. The name is obviously derived from Adams' novels. Today, the company that founded the "42 Network" has more than 15 teaching bases. In the movie "Spider-Man: Parallel Universe", various tricks of "42" also appeared. If you click into the "42" entry on Wikipedia, you will find more interesting allusions. Actually, there are many interesting coincidences about 42, but why these coincidences exist may not be known. For example, in ancient Egyptian mythology, when a person becomes a soul after death, he needs to be judged. The deceased needs to show to 42 judges that he has not committed any of the 42 crimes. In other legends, after the Greeks defeated the Persian Empire, they sent the envoy Phidippides back to Athens from the marathon. The distance covered was about 42.195 kilometers. The distance of the modern marathon is also taken from here (and at that time, There is no such unit as "km"). Tubo has 42 generations of Zanpu, of which the first generation Nie Tri Zanpu came to the throne in about 127 BC. The reign of the last Zanpu, the 42nd Zamprandama, began in 838 AD and ended in 842 AD. The Gutenberg Bible, which was first published in Europe with movable type printing, has 42 lines per page, so it is also called the "forty-two line Bible." On March 6 this year, the "Economist" blog published an article commemorating the 42nd anniversary of the first radio drama series "The Hitchhiker's Guide to the Galaxy" in 1978 (the novel was published only after that). The article read: "Very few people will commemorate the 42nd anniversary." The author just writes casually Many people want to ask, what is the significance of Adams' 42? He succinctly answered this question in the online discussion group: "This is a joke. First, I have to find a simple and short number, and then I decide it is. Binary, thirteen, Tubo Zanpu, etc. All speculations were groundless. I was sitting at the desk, staring at the garden, and thinking, '42 is fine.' Then I typed it out. It's that simple." In binary, 42 is written as 101010, which looks simple and clever. Many fans held a party for this, on October 10, 2010 (10/10/10). But the explanation under the hexadecimal system is not so obvious. You have to answer "How much is six times nine?" to get a clue. In the hexadecimal system, (4 x 13) + 2 = 54. Except for the boring far-fetching of these computer scientists, and some coincidences found in the long river of history, what is special about the number 42 in mathematics? Mathematically unique? 42 has many interesting mathematical properties. Here are a few: The sum of the first three odd exponents of 2: 21 + 23 + 25 = 42. If we take the sum of such n odd powers as a sequence a(n) (that is, 42 = 2(3)), we get the sequence A105281. (OEIS is a website created by mathematician Neil Sloan, which collects all kinds of numbers you can't think of. You can use the first few items to search on it). In the binary system, each item of this sequence is actually to write "10" n times (1010... 10). The general formula of the sequence is a(n) = (2/3) (4n – 1). As n increases, the density of numbers tends to zero. In other words, the numbers in this series, including 42, are actually quite rare. 42 is also the sum of the first two powers of 6: 61 + 62 = 42. The corresponding sequence b(n) here corresponds to A105281 of OEIS. The general formula is b(0) = 0, b(n) = 6b(n – 1) + 6. The density of numbers also tends to 0 at infinity. 42 is a Catalan number. This kind of number is also very rare. There are only 14 Catalan numbers below one million, much less than prime numbers. Euler introduced this concept at that time to answer the question "how many triangles can a convex n-sided shape be decomposed". The first few items in the sequence are 1, 1, 2, 5, 14, 42, 132. . . It can be found in OEIS A000108. The general term formula is c(n) = (2n)! / (N! (n + 1)! ). Like the first two series, the density of numbers also approaches zero infinitely. 42 is also a fairly "practical" number, because any integer between 1 and 42, such as 20, can be decomposed like this: 20=14+6, where 14 and 6 can divide 42 (that is, a factor of 42) , The other numbers from 1 to 42 are the same, they can all be expressed as the sum of different factors of 42. The first few items of such "practical" numbers are: 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60 , 64, 66, 72 (hermes outlet). At present, we do not know the general formula of this sequence. is very interesting, but this does not mean that 42 has any unique meaning in mathematics. Its neighbors 41 and 43 also have many wonderful properties. You only need to search for any number in Wikipedia to find out its various properties. Then how can we tell whether a certain number is interesting or not? Me and two partners: Nicolas Gauvrit, a mathematician and psychologist, and Hector Zenil, a computer scientist, have studied this problem. We also tried to go towards Korotkoff's complexity, but the final result showed that the series included in OEIS are actually mainly from people's preferences. The cube sum of three numbers Computer scientists and mathematicians are sometimes interested in 42, but for them, this is just a small game in their free time, even if they change the number, they can play it. However, a piece of news not long ago attracted their attention. This is the "three cube sum" problem. In this problem, 42 is more challenging than other numbers below 100. The question is this: How to judge whether a number n can be decomposed into the form n = a3 + b3 + c3? How to find such a, b, c? Since abc may be negative, their combinations are endless, unlike the sum of squares. The absolute value of the number decomposed by the sum of squares must be smaller than the original number, so the combination is limited; and given a number, we can definitely judge whether it can be decomposed into the sum of squares. For the sum of cubes, its decomposition may be outrageous, such as 156. The decomposition of this number was discovered in 2007: 156 = 265771108075693 + (−18161093358005) 3 + (−23, 381515025762) 3 Before decomposing, we must first pay attention to a problem, that is, numbers like 9m+4 and 9m+5 cannot be decomposed (like 4, 5, 13, 14, 22, 23). In order to illustrate how difficult it is to find a solution, let us first give two examples, n=1 and n=2. When n=1, it is very simple: 13 + 13 + (–1) 3 = 1 Is there any other decomposition? The answer is yes: 93 + (–6)3 + (–8)3 = 729 + (–216) + (–512) = 1 The solution is more than that. In 1936, the German mathematician Kurt Mahler discovered that for any p, the following formula holds: (9p4) 3 + (3p – 9p4) 3 + (1 – 9p3) 3 = 1 The proof is quite simple, just need to use the binomial expansion learned in middle school: (A + B) 3 = A3 + 3A2B + 3AB2 + B3 For n=2, there are infinitely many solutions. The following formula was created by A in 1908. S. What A. S. Werebrusov discovered: (6p3 + 1) 3 + (1 – 6p3) 3 + (–6p2) 3 = 2 As long as we multiply both sides of the above formula by a perfect cube number (r3), we can get: for any perfect cube number and twice the perfect cube number, there are infinite solutions. For example, 16, it is 23 times of 2, then if p=1, there will be 143 + (–10) 3 + (–12) 3 = 16 When n=3, there are only two known solutions (as of August 2019) 13 + 13 + 13 = 3; 43 + 43 + (–5) 3 = 3 Then naturally we have to ask: Can other numbers be decomposed except for the numbers that have been proved to be indecomposable above? Computer labor In order to answer this question, mathematicians began to verify the numbers 1, 2, 3, 6, 7, 8, 9, 10, 11, 12, 15, 16 besides 9m+4 and 9m+5. . . (A060464). If a solution can be found for the previous numbers, then such a decomposition is likely to be widespread. So far, conscientious computers and computer networks have provided many results for the study of this problem. And in the end we went back to 42. In 2009, two German mathematicians, Andreas-Stephan Elsenhans (Andreas-Stephan Elsenhans) and Jörg Jahnel (Jörg Jahnel) used a method developed by Harvard University’s Noam Ersenhans The method proposed by Noam Elkies in 2000 finds a, b, and c in all the "three cube sum" problems within 1014 for n within 1000. Most n have been answered, except 33, 42, 74, 114, 165, 390, 579, 627, 633, 732, 795, 906, 921, 975. For those within 100, there are only 33, 42 and 74. In 2016, Sander Huisman (now at the University of Twente in the Netherlands), found a 74 solution: (–284650292555885) 3 + (66229832190556) 3 + (28,450105697727) 3 In 2019, Andrew Bookder of the University of Bristol found 33 solutions 88661289752875283 + (–8778405442862239) 3 + (–2736111468807040) 3 So far, Douglas Adams’ 42 is the only unsolved number remaining within 100. If the solution does not exist, 42 is really different. However, computers have not given up, they continue to search for answers. The answer will finally be revealed in 2020. Booker mentioned above and Andrew Sutherland of MIT are the main contributors. Through the charity engine platform, using the equivalent of more than one million hours of computing time, we finally got the result: 42 = (–80538738812075974) 3 + 804357581458175153 + 126021232973356313 165, 795 and 906 have also been announced recently. Now only 114, 390, 579, 627, 633, 732, 921, 975 are left below 1000. It now appears that all numbers except 9m + 4 and 9m + 5 are likely to be decomposed. In 1992, Roger Heath-Brown (Roger Heath-Brown) of Oxford University also proposed a stronger conjecture: He guessed that this decomposition is infinite for every number. However, so far, we are still a long way from proving these conjectures. This question is too difficult. Generally speaking, no algorithm can traverse all possibilities. For example, as early as 1936, Alan Turing (Alan Turing) proved that no algorithm can solve the downtime problem of all computer programs. But now the domain of the problem has reached pure mathematics that is easy to describe. If we can prove the uncertainty of this issue, it will be a big step forward. The number 42 is difficult to understand, but it is not the last step at all!
0 Comments
Leave a Reply. |
ArchivesCategories |