One of my favorite anecdotes about prime numbers comes from Alexander Grothendieck, a 20th-century who was one of the most brilliant mathematicians of the century. According to an accounthe was once asked in an interview to name a prime number. These numbers, which are divisible only by 1 and themselves, form the atoms of number theory, so to speak, and have fascinated mankind for thousands of years.
Grothendieck is said to have replied: “57”. Although the truth of this story is difficult to determine, in nerd circles 57 Grothendieck is known as “number one”, even though it is divisible by 3 and therefore not a number one.
A similar conversation overheard by the mathematician Neil Sloane at a lunch between two of his colleagues, Armand Borel and the late Freeman Dyson, had a more exciting outcome. Borel asked Dyson to name a prime number and, unlike Grothendieck, Dyson gave a number divisible only by 1 and itself: 231 – 1. But that answer did not satisfy Borel. Dyson wanted me to recite all the digits of a large prime number. Dyson was silent, so after a moment, Sloane jumped up and said, “1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6 , 5, 4, 3, 2, 1.”
About supporting science journalism
If you like this article, please consider supporting our award-winning journalism subscribe. By purchasing a subscription, you’re helping to ensure a future of impactful stories about the discoveries and ideas that shape our world.
The number 12,345,678,910,987,654,321 is precisely the first. It consists of 20 digits and is very easy to remember: count to 10 and then count back until you reach 1. But it is not clear whether other prime numbers take the palindromic form starting from 1 and going up to no. n and then down again. Sloane calls the first “memorables,” and they can be presented as 123… (n – 1)n(n – 1) … 321. Because n = 10, you get the number Sloane mentioned. But is there another? nWhat is a prime number result? Dyson, Borel and Sloan must have had a lively conversation about all this.
He was particularly interested in Sloane. He created a database of number sequences in 1964 which eventually formed the spine Online Encyclopedia of Integer Sequences (OEIS)It was launched in 1996. On the OEIS website, experts collect and discuss all kinds of data about the number sequence. Sloane herself was happy to participate in the discussions and repeatedly initiated research questions, and this online activity eventually led to a search for memorable and similar prime numbers.
Is there an unlimited number of first memories?
In 2015 Indian engineer Shyam Sunder Gupta, fascinated by prime numbers since childhood, discovered the number 123 … (n – 1)n(n – 1) … for 321 n = 2,446 is a prime number. He did not publish this in a mathematical journal but he announced the result through a mailing list is used in number theory for these kinds of discoveries. As a result, the first number has 17,350 digits.
“Because prime numbers are so useful in secure communication, large prime numbers that are so easily remembered can be a huge advantage in cryptography,” says Gupta. “That’s why I’m excited about this kind of number one.”
It is not yet known if there are any other first memories. Mathematicians have verified it until all cases n = 60,000; Apart from 10 and 2,446, nothing else was found. But experts suspect that more exist, even if they can’t prove it.
Some argue that there must be an infinite number of primes of this kind. Such “heuristic” arguments assume that the prime numbers are randomly distributed and determine that one type of number (in this case the palindrome 123 … 321) is prime. Although these considerations are not irrefutable proof, they at least provide an incentive for further research. Gupta, for one, is convinced that there should be an infinite number of such palindromes, even if they are rare.
Other types of early memories
September 29, 2015Two months after Gupta shared his result, Sloane posted a call on the number theory mailing list challenging others to find another memorable prime, in which numbers simply add up until a final digit is hit. n: 123… (n – 2)(n – 1)n. To be the first, that number cannot end with an even digit or a 5, which excludes 60 percent of the total. n from the beginning However, even in this case, heuristic arguments suggest that an infinite number of such primes exist.
In response to Sloane’s call, a number of prime number enthusiasts turned on their computers and began systematically searching for a Smarandach prime number, as these known primes are called. After none appear, even for five-digit values in He turned back to Sloane Excellent Internet Mersenne Prime Search the team It’s a collaborative project where volunteers make their computing power available to search for prime numbers. A team from the Mersenne prime search team liked Sloane’s idea, and The search for Smarandache primes was launched under the name Great Smarandache PRPrime search. But after no first number appeared n = 106the project was cancelled.
At first glance, the lack of results seemed surprising. Number 1,234,567,891 is is a prime number, but 12,345,678,910, even, is not. If we take into account the existing restrictions – a prime number cannot be divisible by 2, 3, 4, etc. – we can calculate that among all the numbers of the form 123 no prime number must appear… n from n = 1 to n = 106. At least that’s it Computer scientist Ernst Mayer’s calculation suggests Accordingly, the expected number of Smarandache primes n = 106 It is about 0.6. “So I would like to encourage the world to go ahead and find this missing prime” Sloane said in a Numberphile YouTube video.
Although there has been little progress on this front, Sloan has encouraged people to pursue their curiosity. In 2015, for example, he asked one of his colleagues—computational biologist Serge Batalov—to search for an inverted Smarandache prime number.. He noticed that typing the numbers in descending order (eg 4321) revealed two prime numbers like this: 82818079 … 321 and 3776537764 … 321.
“Can you get one more term? This can be child’s play for you!” he wrote
Batalov’s answer: “Challenge accepted!”
These reverse Smarandache primes inevitably end in 1, meaning far fewer candidates are eliminated from the start. However, to date, Batalov, who has provided much information on similar prime number problems, has not found any new examples of this memorable variety.
Gupta has also helped in the search but to no avail. In 2023 software developer Tyler Busby stated that for the third prime number in the sequence, n It must be greater than 84,300 for n(n – 1) … 321.
How and if the hunt will continue is still unclear. The participants are mostly amateur mathematicians, not professional number theorists. This is because prime numbers of this type do not immediately provide new mathematical information.
However, Sloane does not give up. Today, at the age of 85, he continues to spread his enthusiasm for math and numbers and motivate people to have fun. He certainly doesn’t need to convince Gupta: “I’m still looking for all kinds of big prime numbers that are easy to remember,” says Gupta. And every now and then he finds one.
This article originally appeared the spectrum of science and reproduced with permission.
