My family’s phone number when I was a child was both a palindrome and a prime: 7984897.
My parents had had the number for two decades without noticing it was a palindrome. I still remember my father’s delight when he got off a phone call with a friend: “Doug just said, ‘Hey, I dialed your number backwards and it was still you who answered.’ I never noticed that before!”
A few years later, around 1973, one of the other math nerds at my high school liked to factor seven-digit phone numbers by hand just for fun. I was then taking a programming class—Fortran IV, punch cards—and one of my self-initiated projects was to write a prime factoring program. I got the program to work, and, inspired by my friend, I started factoring various phone numbers. Imagine my own delight when I learned that my home phone number was not only a palindrome but also prime.
Postscript: The reason we hadn’t noticed that 7984897 was a palindrome was because, until around 1970, phone numbers in our area were written and spoken with the telephone exchange name [1]. When I was small, I learned our phone number as “SYcamore 8 4 8 9 7” or “S Y 8 4 8 9 7.” We thought of the first two digits as letters, not as numbers.
Second postscript: I lost contact with that prime-factoring friend after high school. I see now that she went on to earn a Ph.D. in mathematics, specialized in number theory, and had an Erdős number of 1. In 1985, she published a paper titled “How Often Is the Number of Divisors of n a Divisor of n?” [2]. She died two years ago, at the age of sixty-six [3].
[1] https://en.wikipedia.org/wiki/Telephone_exchange_names
[2] https://www.sciencedirect.com/science/article/pii/0022314X85...
[3] https://www.legacy.com/us/obituaries/legacyremembers/claudia...
> Since prime numbers are very useful in secure communication, such easy-to-remember large prime numbers can be of great advantage in cryptography
What's the use of notable prime numbers in cryptography? My understanding is that a lot of cryptography relies on secret prime numbers, so choosing a notable/memorable prime number is like choosing 1234 as your PIN. Are there places that need a prime that's arbitrary, large, and public?
https://en.wikipedia.org/wiki/Belphegor%27s_prime
"666" with 13 0's on either side and 1's on the ends.
As soon as I read the title of this post, the anecdote about the Grothendieck prime came to mind. Sure enough, the article kicks off with that very story! The article also links to https://www.ams.org/notices/200410/fea-grothendieck-part2.pd... which has an account of this anecdote. But the article does not reproduce the anecdote as stated in the linked document. So allow me to share it here as I've always found it quite amusing:
> One striking characteristic of Grothendieck’s mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called “Grothendieck prime”. In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.”
Doesn't take very much searching to find this pretty nifty palindrome prime:
3,212,123 (the 333rd palindrome prime)
Interestingly, there are no four digit palindrome primes because they would be divisible by 11. This is obvious in retrospect but I found this fact by giving NotebookLM a big list of palindrome primes (just to see what it could possibly say about it over a podcast).
For the curious, here's a small set of the palindrome primes: http://brainplex.net/pprimes.txt
The format is x. y. z. n signifying the x-th prime#, y-th palindrome#, z-th palindrome-prime#, and the number (n). [Starting from 2]
Since prime numbers are very useful in secure communication, such easy-to-remember large prime numbers can be of great advantage in cryptography,
That's nonsense. I'm sure there thinking of RSA, but that needs secret prime numbers. So easy-to-remember is pretty much the opposite of one want. Also they are way to big. 2048 bit RSA needs two 300 digit prime numbers.
The title of the Scientific American article is "These Prime Numbers Are So Memorable That People Hunt for Them", which matches the content much better than the title above.
A few other memorable primes:
https://math.stackexchange.com/questions/2420488/what-is-tri...
888888888888888888888888888888
888888888888888888888888888888
888888888888888888888888888888
888111111111111111111111111888
888111111111111111111111111888
888111111811111111118111111888
888111118811111111118811111888
888111188811111111118881111888
888111188811111111118881111888
888111888811111111118888111888
888111888881111111188888111888
888111888888111111888888111888
888111888888888888888888111888
888111888888888888888888111888
888111888888888888888888111888
888811188888888888888881118888
188811188888888888888881118881
188881118888888888888811188881
118888111888888888888111888811
111888811118888888811118888111
111188881111111111111188881111
111118888111111111111888811111
111111888811111111118888111111
111111188881111111188881111111
111111118888811118888811111111
111111111888881188888111111111
111111111118888888811111111111
111111111111888888111111111111
111111111111118811111111111111
111111111111111111111111111111
062100000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000000
000000000000000000000000000001
777777777777777777777777777777777777777
777777777777777777777777777777777777777
777777777777777777777777777777777777777
777777777777777777777777777777777777777
111111111111111111111111111111111111111
111111111111111111111111111111111111111
188888888118888888811188888811188888811
188111118818811111881881111881881111881
188111118818811111881881111111881111111
188888888118888888811881111111881118888
188111111118811111111881111111881111881
188111111118811111111881111881881111881
188111111118811111111188888811188888811
111111111111111111111111111111111111111
111111111111111111111111111111111111111
333333333333333333333333333333333333333
20181111111111111111111111111111111111
11111111111111111166111111111111111111
11111111111111111868011111111111111111
11111111111111118886301111111111111111
11111111111111168863586111111111111111
11111111111111803608088361111111111111
11111111111193386838898668111111111111
11111111111111163508800111111111111111
11111111111111806560885611111111111111
11111111111118630808083861111111111111
11111111111585688085086853511111111111
11111111116355560388530533881111111111
11111111506383308388080803858311111111
11111183585588536538563360080880111111
11111111111118383588055585111111111111
11111111111568838588536853611111111111
11111111118830583888838553631111111111
11111111808885338530655586888811111111
11111183886860888066566368806366111111
11115385585036885386888980683008381111
11055880566883886086806355803583885511
11111111111111111685311111111111111111
11111111111111111863311111111111111111
11111111111111111035611111111111111111Since divisibility by 2 and 5 is such a problem, why not look for memorable numbers in prime base, like base 7 or base 11?
ChatGPT o1: https://chatgpt.com/share/678feedb-0b2c-8001-bd77-4e574502e4...
> Thought about large prime check for 3m 52s: "Despite its interesting pattern of digits, 12,345,678,910,987,654,321 is definitely not prime. It is a large composite number with no small prime factors."
Feels like this Online Encyclopedia of Integer Sequences (OEIS) would be a good candidate for a hallucination benchmark...
On the topic of palindromic numbers, I remember being fascinated as a kid with the fact that if you square the number formed by repeating the digit 1 between 1 and 9 times (e.g. 111,111^2) you get a palindrome of the form 123...n...321 with n being the number of 1s you squared.
The article talks about a very similar number: 2^31-1, which is 12345678910987654321, whereas 1111111111^2 is 12345678900987654321
Not quite the same, but this reminds me of bitcoin, where miners are on the hunt for SHA hashes that start with a bunch of zeroes in a row (which one could say is memorable/unusual)
Maybe there's a prime number that makes a mildly interesting picture when rendered in base-2 in a 8*8 grid.
Should somebody spend time looking at all the primes that fit in the grid? Absolutely not.
Reminds me the demonstration that all whole numbers are interesting in a way or another. Being memorable in this case is not so much about memory but about having an easy to notice pattern of digits, or a clear trivial algorithm to build them.
34567876543
333 2 111 2 333
1111 4 7 4 1111
35753 3 35753
At one time, in university, I wrote a tool to aesthetically score primes.
I there any more l33t prime than 31337?
> Sloane calls them “memorable” primes
Excluding 11 seems arbitrary here.
...in decimal.
https://t5k.org/notes/words.html points out that "When we work in base 36 all the letters are used - hence all words are numbers." Primes can be especially memorable in base 36. "Did," "nun," and "pop" are base-36 primes, as is "primetest" and many others.
If you were around in the 80's and 90's you might have already memorized the prime 8675309 (https://en.wikipedia.org/wiki/867-5309/Jenny). It's also a twin prime, so you can add 2 to get another prime (8675311).