A delightful story about prime numbers features Alexander Grothendieck, a leading mathematician of the 20th century. It’s said that during a discussion, he was asked to identify a prime number. Prime numbers, which are divisible only by 1 and themselves, are foundational to number theory and have captivated people for millennia.

Grothendieck reportedly answered, “57.” While the authenticity of this tale is uncertain, 57 has humorously been dubbed the Grothendieck “prime number” in mathematical circles, despite being divisible by 3 and not actually a prime.

Another intriguing incident was recounted by mathematician Neil Sloane about a conversation he witnessed between Armand Borel and the late Freeman Dyson. Borel challenged Dyson to name a prime number, and Dyson responded with 2³¹ – 1, a true prime. Unsatisfied, Borel pressed for the recitation of a long prime number. Dyson remained silent, prompting Sloane to humorously chime in with, “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 prime, made up of 20 digits and is remarkably easy to recall: just count up to 10 and back down to 1. However, it remains to be seen if other primes exist that mimic this palindromic pattern. Sloane refers to these as “memorable” primes, and they can be represented as 123 … (n – 1)n(n – 1) … 321. For n = 10, this is the number Sloane mentioned. Whether other values of n yield primes remains an open question discussed by Dyson, Borel, and Sloane.

Sloane, deeply intrigued by this, founded a database of number sequences in 1964 that ultimately became the foundation for the Online Encyclopedia of Integer Sequences (OEIS) launched in 1996. On the OEIS website, experts gather various data on number sequences and engage in discussions, with Sloane frequently initiating investigative queries that spurred the search for memorable primes and similar figures.

Are There Endless Memorable Primes?

In 2015, Indian engineer Shyam Sunder Gupta, a lifelong prime number enthusiast, discovered that the number 123 … (n – 1)n(n – 1) … 321 for n = 2,446 is a prime. Rather than publishing in a journal, he announced his findings via a number theory mailing list. This prime has 17,350 digits.

Gupta points out that such easily remembered large primes could be very beneficial in secure communications and cryptography. This fuels his passion for these special prime numbers.

Whether more memorable primes exist is still unknown. Mathematicians have tested up to n = 60,000; so far, only 10 and 2,446 have been prime. However, there is speculation that more could exist, though unproven.

Some suggest that an infinite number of such primes should exist, based on heuristic assumptions about the random distribution of prime numbers along the numerical spectrum and the likelihood of certain numbers being prime. While not definitive proof, these theories encourage further investigation, and Gupta remains hopeful about discovering more of these prime palindromes.

Varieties of Memorable Primes

On September 29, 2015, following Gupta’s discovery, Sloane challenged the number theory community to find another type of memorable prime where numbers ascend to a final digit n: 123 … (n – 2)(n – 1)n. For a number to be prime, it cannot end in an even number or a 5, which automatically rules out 60 percent of potential n values. Despite this, heuristic theories suggest an endless number of such primes might exist.

In response, enthusiasts began using their computers to search systematically for a Smarandache prime, as these primes are named. After none were found even up to five-digit n values, the search was escalated to the Great Internet Mersenne Prime Search team. This collaborative project uses volunteer computing power to hunt for primes. Despite high hopes, no Smarandache primes were discovered even up to n = 10⁶, leading to the project’s discontinuation.

Initially, it was surprising to see no results. The number 1,234,567,891 is a prime, but 12,345,678,910 is not. Taking into account that a prime number cannot be divisible by 2, 3, 4, etc., it was estimated that no prime would occur among numbers in the form 123 … n up to n = 10⁶, according to computer scientist Ernst Mayer. The expected number of Smarandache primes up to that point was about 0.6. “So I would like to encourage the world to keep searching for this elusive prime,” Sloane expressed in a Numberphile YouTube video.

Despite limited progress, Sloane continues to inspire curiosity and exploration. In 2015, he encouraged computational biologist Serge Batalov to look for a reverse Smarandache prime, where numbers are written in descending order. So far, two such primes have been discovered: 82818079 … 321 and 3776537764 … 321.

“Can you find another? This might be easy for you!” he suggested.

Batalov accepted the challenge enthusiastically.

These reverse Smarandache primes always end in 1, reducing the number of disqualified candidates. However, to date, Batalov, despite his contributions to similar prime number challenges, has yet to discover new examples of this type.

Gupta also continued his search without success. In 2023, software developer Tyler Busby noted that for the third prime in the sequence, n must exceed 84,300 for n(n – 1) … 321.

The future of this quest remains uncertain, with mostly amateur mathematicians involved rather than professional number theorists, as these types of prime numbers don’t necessarily yield new mathematical insights.

Yet, Sloane, now 85, remains undeterred, continuing to foster a love for mathematics and encouraging others to enjoy the thrill of discovery. Gupta echoes this sentiment, constantly on the lookout for large, memorable prime numbers, occasionally finding success.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.