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 is prime is approximately . 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.