One of my favorite anecdotes about prime numbers concerns Alexander Grothendieck, one of the greatest mathematicians of the 20th century. According to one report, he was once asked to name a prime number during a conversation. These numbers, which are only divisible by 1 and themselves, form the atom of number theory, so to speak, and have fascinated humanity for thousands of years.
Grothendieck is said to have replied, “57.” Although it is difficult to determine the truth of this story, 57 has since become known in geek circles as Grothendieck’s “prime”, even though it is not a prime number because it is divisible by 3.
A similar conversation that mathematician Neil Sloan overheard over a meal with his colleagues Armand Borel and the late Freeman Dyson had even more exciting results. Borel asked Dyson to name a prime number, but unlike Grothendieck, Dyson provided a number that was only divisible by 1 and itself.31 – 1. But that answer did not satisfy Borel. He wanted Dyson to recite all the digits of large prime numbers. Dyson went silent, and after a while Sloane cut in. “1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1.”
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The number 12,345,678,910,987,654,321 is certainly a prime number. It consists of 20 digits and is very easy to remember. Count to 10, then count backwards again until you reach 1. However, it is unclear whether other prime numbers take the form of a palindrome, starting at 1 and increasing in numerical order. n Then it descends again. Sloan calls them “memorable” primes, which can be represented as 123…(n – 1)n(n – 1) … 321. n = 10, giving us the number Sloane mentioned. But maybe there are others? nWhich result is a prime number? Dyson, Borel, and Sloan must have had a lively conversation about all this.
Sloane was particularly interested. He created a database of number sequences in 1964, which eventually formed the backbone of the Online Encyclopedia of Integer Sequences (OEIS), launched in 1996. On the OEIS website, experts compile and discuss all kinds of facts about number sequences. Sloan himself happily joined the discussion and began asking repeated questions about the research. This online activity ultimately led to a memorable search for similar prime numbers.
Are there an infinite number of memorable prime numbers?
In 2015, Shyam Sundar Gupta, an Indian engineer who has been fascinated by prime numbers since childhood, discovered the number 123…n – 1)n(n – 1) … if 321 n = 2,446 is a prime number. He did not publish this in a mathematical journal, but announced his results through a mailing list used for this kind of discovery in number theory. The resulting prime number has 17,350 digits.
“These large, easy-to-remember primes are a big advantage in cryptography, because prime numbers are very useful for secure communications,” says Gupta. “That’s why I’m so passionate about this kind of prime number.”
It is still unclear whether there are any other memorable prime numbers. Mathematicians have verified all previous cases. n = 60,000; Apart from 10 and 2,446, nothing else was found. But experts suspect there are more, even if they can’t prove it.
Some argue that there must be an infinite number of prime numbers of this type. Such “heuristic” arguments assume that the prime numbers are randomly distributed on the number line and calculate how likely it is that a particular type of number (in this case the palindromes 123…321) is prime. Determine the degree. Although these considerations are not incontrovertible evidence, they at least provide motivation for further research. For example, Mr. Gupta believes that there must be an infinite number of such palindromes, even if they are rare.
Other kinds of memorable primes
On September 29, 2015, about two months after Mr. Gupta shared his results, Mr. Sloan posted a call to action on the Number Theory mailing list, calling for a different kind in which numbers simply ascend until they reach the last digit. challenged others to find a memorable prime number for . n: one two three … (n – 2)(n – 1)n. To be a prime number, such a number cannot end with an even number of digits or 5. This excludes 60 percent of the total. n From the beginning. But even in this case, heuristic arguments suggest that there are an infinite number of such primes.
In response to Sloan’s call, some prime number enthusiasts fired up their computers and began a systematic search for smarandash primes (as these particular primes are called). After nothing was displayed even with a 5-digit value, yeah, Mr. Sloan consulted the Great Internet Mersenne Prime Search team. This is a collaborative project where volunteers can use their computing power to search for prime numbers. A group from the Mersenne Prime Search Team liked Sloan’s idea, and the search for Semarandas primes was launched under the name Great Semarandas PRPrime Search. But after no prime numbers appear, n = 106the project was abandoned.
At first glance, the lack of results seemed surprising. Number 1,234,567,891 teeth Although it is a prime number, 12,345,678,910 is not an even number. Considering the limitations that exist (prime numbers are not divisible by 2, 3, 4, etc.), we can deduce that a prime number cannot occur among all numbers of the form 123 …. n from n = 1 ~ n = 106. At least that’s what computer scientist Ernst Mayer’s calculations suggest. According to this, the expected number of smarandash primes is at most n = 106 It’s about 0.6. “So I would encourage the world to keep moving forward to find this missing prime number,” Sloan said in a Numberphile YouTube video.
Although little progress has been made in this regard, Sloan has encouraged people to remain curious. In 2015, for example, he encouraged one of his colleagues, the computational biologist Serge Batalov, to look for inverse Smarandash primes. He pointed out that writing out numbers in descending order (for example, 4321) has so far revealed two prime numbers: 82818079 … 321 and 3776537764 … 321.
“Can I have another term? This could be child’s play for you!” he wrote.
Batalov replied: “Challenge accepted!”
All of these inverse smarandash primes always end in 1. This means that far fewer candidates are eliminated from the start. Batalov has provided many insights into similar prime number problems, but so far he has not discovered any new examples of this memorable variety.
Mr. Gupta also cooperated in the search, but to no avail. In 2023, software developer Tyler Buzbee said of the third prime number in the sequence: n Must be greater than 84,300 n(n – 1) … 321.
It is still unclear how and if hunting will continue. Participants are primarily amateur mathematicians, not professional number theorists. This is because prime numbers of this type do not immediately provide new mathematical insights.
Still, Sloan doesn’t give up. Now 85 years old, he continues to spread his enthusiasm for mathematics and numbers and inspire others to enjoy them. There is certainly no need to convince Mr. Gupta. “I’m looking for all kinds of large prime numbers that are easy to remember,” says Gupta. And sometimes he finds it.
This article was first published Wissenschaft spectrum Reprinted with permission.
