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The Greek mathematician Euclid proved very well that K. a. Around 300, there are many infinite prime numbers. But it was the British mathematician Christian Lawson-Perfect who recently invented the computer game. “Is this the first?”
Launched five years ago, the game surpassed three million attempts on July 16 — or rather ran 2,999,999. Hacker News message He created about 100,000 attempts.
The goal of the game is to sort as many “first” or “not first” numbers as possible in 60 hours (originally Lawson-Perfect) describe is on Aperiodical, mathematics blog that is the creator and editor).
A prime number is an integer with exactly two divisors, 1 and the same.
“It’s very easy, but it’s very hard,” says Lawson-Perfect, who works in the e-learning unit at the School of Mathematics and Statistics at Newcastle University. He created the game in his spare time, but it was useful at work: Lawson-Perfect writes electronic assessment software (systems that assess learning). “The system I do is designed to randomly answer and answer the math question, which automatically marks and gives feedback on it,” he says. “You can see the first games as a kind of assessment” – used in outreach sessions in schools.
It made the game a bit easier with keyboard shortcuts — the y and n keys click on the corresponding yes-no buttons on the screen — saving you time to move the mouse.
Take a look:
Primality verification algorithms
Prime numbers have practical utility in computing, for example, with error-correcting codes and encryption. But although the first factorization is hard (its value in encrypting from there), the verification of dominance is easier, if not difficult. The German mathematician won the Fields Medal Alexander Grothendieck wrong wrong 57 for the former (“Grothendieck prime”). Lawson-Perfect when has analyzed the game dataHe saw that some numbers showed “Grothendieckyness.” The number most often solved with a common number was 51, followed by 57, 87, 91, 119, and 133 — the enemy of Lawson-Perfect (who also invented a useful service for verifying superiority: https://isthisprime.com/2).
The most minimalist algorithm for verifying the dominance of a number is the division test: divide the number by each number until its square root (the product of two numbers more than the square root would be greater than the number of cases).
However, this naive method is not very effective, nor are other techniques invented over the centuries — as the German mathematician Car Friedrich Gauss saw in 1801, “they require unbearable work even for the relentless calculator.”
The game-coded Lawson-Perfect algorithm is based on the Miller-Rabin dominance test (a very effective seventeenth-century method but not iron). “Fermat’s little theorem”). The Miller-Rabin test works amazingly well. As for Lawson-Perfect, it’s “basically magic” – “I don’t understand how it works, but I’m sure I would spend time exploring it in more depth,” he says.
Because the test uses random, it produces a probabilistic result. This means that sometimes the test is a lie. “There’s a chance to discover an impostor who’s trying to get past the compound number like before,” says Carl Pomerance, a mathematician and author of the book at Dartmouth College. Prime numbers: a computational approach. There are a trillion chances of an imposition slipping from an algorithm’s quick verification mechanism, but the test is “fairly secure”.
But when it comes to clever algorithms for verifying dominance, the Miller-Rabin test is “the tip of the iceberg,” Pomerance says. It is noteworthy that 19 years ago, three computer scientists — Manindra Agrawal, Neeraj Kayal, and Nitin Saxena — all at the Indian Institute of Technology in Kanpur — reported AKS dominance test (again based on Fermat’s method), which in the end proved to be a prime number, which proved to be random and (theoretically, at least) at an impressive rate. Alas, theoretically fast does not always come to life, so the AKS test is not useful for practical purposes.
Unofficial world record
But practicality is not always an issue. From time to time Lawson-Perfect receives emails from people who are eager to share high scores in the game. A player recently reported 60 premiums in 60 seconds, but the record is 127. Leadership does not track high scores; he knows that there are some pitfalls with computer-assisted attempts to generate points in data.
Score 127 was obtained by Ravi Fernando, a graduate in mathematics from the University of California, Berkeley published the result in July 2020. He still has his personal best and, he believes, is an “unofficial world record”.
Since last summer, Fernando hasn’t played many games with the default settings, but has tried custom settings, picking higher numbers and accepting longer time limits – he scored 240 points with a five-minute limit. “That made a lot of inventions, because the numbers went into a high four-digit range and I only learned the premiums of the 3,000s,” he says. “I think some would argue that this is an exaggeration.”
Fernando’s research is in algebraic geometry, which includes premiums to some extent. However, he says, “my research has more to do with why I started gambling than why I started gambling” (I started my PhD in 2014). Also, overcoming the 127 numbers would be very difficult. And, he says, “it’s right to stop at a record of prime numbers.”
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