Prime numbers so memorable that people hunt for them
105 comments
·January 18, 2025tkgally
tkgally
> In 1985, she published a paper titled “How Often Is the Number of Divisors of n a Divisor of n?”
Claudia Spiro seems to have remained actively interested in prime numbers into her sixties. In 2017, she published a paper titled “On three consecutive prime-gaps”:
https://projecteuclid.org/journals/rocky-mountain-journal-of...
dhosek
I thought everybody factors phone numbers. I also factor the odometer reading in my car while driving.
noman-land
How do you do mental factoring?
dhosek
I don’t usually keep the factors, but I have a variety of techniques. Putting aside the trivial cases of divisibility by 2, 3, 5, 11¹ and 37², a lot of it comes down to find ways to make the numbers smaller, so, for example. my current odometer reading is 13,857. To test for divisibility by 7, I can turn that into 14,000-13852=143 which I can tell at a glance isn’t divisible by 7. It is divisible by 3, so I can reduce it to 4619 which isn’t divisible by 3 and I can also tell it’s not divisible by 11. I can also rule out 19 at a glance. To check 13, I might do my right-side divisibility test where I start by subtracting 39 from the right, which gives 458. I could continue with that, but taking 39 from the left gets me to a small enough number faster of 68 which isn’t divisible by 13. Right-side divisibility for 17 takes 119 from the right leaving 45 which isn’t divisible by 17. For 23, I can do a left-side removal of 46 to rule that out. 29, go right to bring it to 459 and then 43, not divisible by 29. For 31, I’ll do right side to get 434 and then 31 so 4619=149×31 and 149 is prime and I’m done.
⸻
1. To check for divisibility by 11, subtract the sum of even numbered digits from the sub of odd numbered digits. If 11∣n, then you’ll have a multiple of 11. E.g., for 13,857, we compare (3+5)-(1+8+7)=-8 which is not a multiple of 11.³
2. To check for divisibility by 111, we take advantage of the fact that 3×37=111, 27×37=999 and thus 1000≡1(mod 37) and we can then add up the digits in groups of 3, and pull out the most convenient multiple of 111 to see if we have 0, 37 or 74. E.g., with our example about, 13+852=865-888=-23 which is not a multiple of 37.
3. As an added bonus that number is the remainder when dividing by 11. Similarly the number I get with the check in footnote 2 is the remainder when dividing by 37.
tkgally
Follow-up: My formal study of programming stopped with that Fortran IV class a half century ago, but LLMs now delude me into thinking I can program. I just had Claude write me a Python program to list all of the seven-digit numbers that are both primes and palindromes. It found 668: 1003001, 1008001, 1022201, 1028201, 1035301, ...
jodrellblank
I wondered if we map English letters to digits (a=0, b=1, c=2, ... , z=25) is there anything which is both a palindrome in its English form and in its number form, and prime? I took a list of palindromes from Wikipedia[1] and tested them, and yes:
deified 3485843
bib 181
did 383
I note that none of them use letters which map to double-digits (k=10, l=11, m=12, etc.); none of the palindrome sentences make palindromic numbers (ignoring case, stripping all punctuation); could there be one which does?
[1] https://en.wiktionary.org/wiki/Appendix:English_palindromes
tkgally
How about representing the words as numbers in base 26? That would preserve the palidromicity for both the word and the number, and primality would not be affected by the base representation.
Might not be as much fun, though.
stevelosh
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).
nlh
My other favorite fun fact about this number (other than this new prime info which I am excited to have learned) is that in almost every store I’ve tried it, someone has used that (along with a local area code) as the phone number for a store loyalty card.
I’m a Bay Area guy, so if you’re ever at Safeway and need to get the discount without giving up your personal info, 415-867-5309 has got ya covered ;)
pwg
> someone has used that (along with a local area code) as the phone number for a store loyalty card.
Usually because for far too long, noisy retailers wanted a "phone number" upon checkout (even if one was paying cash -- Radio Shack was an especially bad one back in the day). For those who didn't want to get yet more telemarketing calls, repeating "Jenny's number" [1] from the song was a way to "just buy" whatever it was you wanted. The minimum wage cashier didn't care, but the cash register demanded "a number". So giving the cashier Jenny's number worked.
This has largely faded now that they can track everyone via one's credit card numbers.
noahjk
Does contactless payment help at all? I know it uses a different card number, but I’m not sure if it’s a rotating number.
fancy_pantser
You can use the number with your local area code just about anywhere at the pump to get a gas discount as well (a common loyalty reward program benefit).
jedberg
I was around in the 80s, but this is awesome new information!
out-of-ideas
lol i didnt realize this was a prime number but i re-use this number any time i need a fake phone number in some sample/example data (im pretty certain nobody gets the reference, or takes the time to read it)
freedomben
I do the same thing! Downside is every time I read my tests I get that song stuck in my head. Upside is it's a joy for some people when they discover it in the test
lehi
https://en.wikipedia.org/wiki/Belphegor%27s_prime
"666" with 13 0's on either side and 1's on the ends.
fuzzythinker
"on both sides" because "on either side" to me meant it may be duo of 1-13zeros-6661 and 1666-13zeros-1.
More for those who don't click the link, other Belphegor primes numbers are with the following number of zeros in both ends (and 1 to cap off the ends): 0, 13, 42, 506, 608, 2472, 2623, maybe more.
sdwr
"to either side" or "on either side" commonly means "on both sides"
"Either" has two meanings:
- verb-wise, it separates different options (you can have either X or Y)
- noun-wise, it refers to two similar groups (there was no light on either side of the bridge, or, conversely, the bridge was lit on either side)
quuxplusone
Indeed. "On either side the river lie / Long fields of barley and of rye" —Tennyson
jjtheblunt
(Native speaker) i read either in the sense of logical or, so one side alone (tegardless of which side) or both sides at once.
Interesting how varied the ohrasing can be read, though!
TeMPOraL
> Belphegor (or Baal Peor, Hebrew: בַּעַל-פְּעוֹר baʿal-pəʿōr – “Lord of the Gap”) is, in the Abrahamic religions, a demon associated with one of the seven deadly sins. According to religious tradition, he helps people make discoveries. He seduces people by proposing incredible inventions that will make them rich.
Huh. Would feel right at home in our industry.
> According to some demonologists from the 17th century, his powers are strongest in April.
Any demo days or other significant VC stuff happening in April?
> The German bishop and witch hunter, Peter Binsfeld (ca. 1540–ca.1600) wrote that Belphegor tempts through laziness. According to Binsfeld's Classification of Demons, Belphegor is the main demon of the deadly sin known as sloth in the Christian tradition. The anonymous author of the Lollard tract The Lanterne of Light, however, believed Belphegor to embody the sin of gluttony rather than sloth.
Yeah, hits too close to home.
irrational
Can’t spell “demon” without “demo”. Cue the church lady.
alphan0n
Startup > trapt (archaic) us
WorldMaker
> Any demo days or other significant VC stuff happening in April?
Lots of tech companies plan elaborate demos for April 1st, for some foolish reason. It certainly gets very busy on HN keeping up.
miki123211
How was this never mentioned in Unsong? Not a single time?
TeMPOraL
IDK, I guess Scott Alexander didn't do his research thoroughly enough. Still, UNSONG is already pretty much a fractal of references and callouts to such things.
On that note, how is it I've never seen anyone connecting the famous "God of the gaps"[0] with a demon literally named "Lord of the Gap"?
(In case no one really did, let history and search engines mark this comment as the first.)
--
gpderetta
Makes sense, with laziness being one of the three virtues of a great programmer.
vdjskshi
Sounds like the patron saint of LLMs
idiotsecant
It also works with no zeros, or all sorts of other number of zeros. Dude basically just added zeros until the number got cooler.
shagie
The palindromic Belphegor numbers https://oeis.org/A232449
Indices of Belphegor primes: numbers k such that the decimal number https://oeis.org/A232448
yapyap
wow, evil pi.
very interesting, thanks for sharing.
susam
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.”
zellyn
One of my pet hobbies is trying to figure out the least prime prime number and most prime composite numbers under 100.
My votes are 61 or 89 for least prime-seeming primes and 87 and --yep-- 57 for more prime-seeming composites.
im3w1l
I'm gonna vote 91, since it has large divisors that can't be seen at a glance. 57 and 87 fall apart if you remember that 60 and 90 are divisible by 3.
expensive_news
I really enjoy this “proof” that the most prime-seeming composite is 91
xigoi
I once wrote in a Math Olympiad solution that 87 is prime. Not my brightest moment.
bhasi
But it's not prime - what am I missing? Why is this anecdote significant?
eigenket
The point is that Grothendieck, easily one of the greatest mathematicians of all time, who regularly proved deep and fundamental facts about prime numbers, cared so little about particular numbers that he accidentally gave an easy to see non-prime as an example of a prime.
He was used to working on completely different levels of abstraction, so when faced with concrete numbers he could easily make a mistake that a school-child (or hacker news commenter) could spot.
bongodongobob
Yeah I don't get it either.
monadINtop
[dead]
hatthew
> 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?
arh68
I believe they're talking about something like ECC
> To use ECC, all parties must agree on all the elements defining the elliptic curve, that is, the domain parameters of the scheme. The size of the field used is typically either prime (and denoted as p) or is a power of two
like "25519"
> An EdDSA signature scheme is a choice: ... of finite field F q over odd prime power q ... Ed25519 is the EdDSA signature scheme where q = 2^255 - 19
[1] https://en.wikipedia.org/wiki/Elliptic-curve_cryptography#Do...
ramchip
A lot of public-key cryptography is based on finite fields, and the integers modulo N combined with addition and multiplication form a finite field iff N is prime. This brings some extremely useful properties e.g. every non-zero element has a multiplicative inverse. N must also be public so people can compute things (e.g. to verify a signature), and big to resist brute force attacks.
pwg
RSA [1] is directly based on the difficulty of factoring very large numbers where the factors of the number are also prime numbers.
xigency
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]
DerekL
> Interestingly, there are no four digit palindrome primes because they would be divisible by 11.
In fact, this holds for any even number of digits.
schneehertz
11 is an even-digit palindrome and is a prime number
DerekL
I meant that every palindrome number with an even number of digits is divisible by 11, not that they're not prime.
oneshtein
11111111111111111111111 is prime (1©23)
quuxplusone
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
11111111111111111035611111111111111111jmward01
These are great! I wonder if Carl Sagan knew about them when writing Contact. The movie doesn't go into the part of the book that is relevant here (trying to avoid spoilers but if you read the book you know!)
quuxplusone
Here's a previous HN submission about finding Waldo in pi (spoiler: only by cheating significantly re what counts as "Waldo"): https://news.ycombinator.com/item?id=30872676
I googled around trying to figure out what year James McKee created the Trinity Hall prime. The internet is (IMO) presenting it mainly as some kind of Wonder of the Ancient World — with the date of creation conveniently filed off. The first post below claims that the year McKee left Cambridge and created the prime was 1996. It seems to have hit peak internet presence only in the 2010s, though, so I wish there were an authoritative source to confirm (or deny) the 1996 date.
https://www.bradyharanblog.com/blog/artistic-prime-numbers
https://www.futilitycloset.com/2017/09/10/trinity-hall-prime...
https://www.futilitycloset.com/2020/01/12/more-prime-images/
noman-land
These are so cool.
slwvx
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.
nurumaik
Since divisibility by 2 and 5 is such a problem, why not look for memorable numbers in prime base, like base 7 or base 11?
elcomet
I can't tell if this is a joke if if you're serious
euroderf
Why do we care about base 10 ? Because we have five digits per appendage ? BFD. Accident of evolution.
What about palindromes in binary ? That's about as close to a mathematical ideal as we could get. Yes?
Let's see. decimal 11 = binary 1011, its palindrome = 1101 = decimal 13, GOLD!
Retr0id
If we allow non-decimal bases, (2^n)-1 works for a lot of memorable values of n (e.g. 2, 3, 5, 7... and 31, per the article), or some less memorable but very long values of n, like 136279841
They're all technically palindromes in base-2.
dchichkov
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...
scotty79
I think firmly marrying llms with symbolic math calculator/database, so they can check things they don't really know "by heart" would go a long way towards making them seem smart.
I really hope Wolfram is working on LLM that is trying to learn what it means to be WolframAlpha user.
bongodongobob
Can we stop with the "haha llms can't do math" nonsense? You'll one shot it every time if you tell it to use Python. You're holding it wrong.
dchichkov
Sorry, but this was ChatGPT/o1 with access to code execution (Python) and it used almost 4 minutes to do reasoning. It had done a few checks with smaller numbers, all of which had failed. And it proceeded to make a wrong conclusion (with high confidence).
bongodongobob
Of course it failed. Tell it to write a program.
lpolovets
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)
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...