Posted by: Dave Richeson | September 3, 2009

## More prime phone numbers

Recently I noticed that Jenny’s phone number 8675309 (867-5309) had some interesting number theoretic properties.

So, this evening I decided to punch my own phone number into Wolfram|Alpha to see what I’d find. Amazingly, both my 7-digit phone number and my full 10-digit phone number are prime numbers! That’s so excellent. (Unfortunately, putting a 1 in front of it makes it composite.)

According to the prime number theorem, the chance that a number $N$ is prime is approximately $1/\ln(N)$. Thus, the chance that my 7-digit phone number is prime is approximatly 7% and the chance that my 10-digit number is prime is about 4.4%. So the chance that they are both prime is approximately 0.3% (assuming they are independent, which I guess they aren’t since knowing that one is prime implies that the other one is at least odd, so maybe the chance of both being prime is more like 0.6%…?)

Actually, that got me to thinking, what is the longest prime number with the property that if we drop the left-most digits one-at-a-time, we always have primes? I looked at this list of primes and found 96823. We see that 3, 23, 823, 6823, and 96823 are all primes. My guess is that we can find such a number of arbitrary length.

Speaking of primes, this whole thread began when I posted a Nova ScienceNow video about the twin prime conjecture. One of my colleagues pointed out that the filmmaker erred by implying that 1 is a prime number. See the screenshot below with the primes colored red. Oops.

## Responses

1. Sounds like you’re talking about Truncatable primes. Surprisingly enough, there are only finitely many of them. The biggest is 357686312646216567629137.

By the way, my (7 digit) phone number is the hypotenuse of two different pythagorean triples! (But not prime.)

Love the blog-

• Hi Chris, great to hear from you! I hope you are enjoying your new job. We miss not having you here in central PA.

Thanks for the tip about truncatable primes. Cool that it has a name and that the set is finite. When I was writing the post I meant to add that zeros were not allowed (as the wikipedia article says), otherwise you could get truncatable primes like 100003.

I also wondered about dropping digits from the right or the left which, according to your link, also exist (15 of them), 739397 being the largest—they are called two-sided primes.

Carrying this discussion to its logical conclusion we could ask if there is a number with the property that if we drop digits one at a time from anywhere in the number, the resulting numbers are prime. It looks like 2, 3, 5, 7, 23, 37, 53, 73, 317 is the complete list.

2. Technically, 317 does not fit your requirement of dropping digits from anywhere. Drop the 3 and the 7, and we lose.

• Oops. Thanks, you are correct.

3. I wrote javascript app to figure out the prime factors of numbers… it does it instantly as you type perfect, for playing with truncatable primes. http://www.coderenaissance.com/2011/06/finding-prime-factors-in-javascript.html. Hope it helps.

Thanks for the great site.