# Prime telephone numbers

Michael Lugo pointed out that the telephone number 867-5309 is prime and may be the largest prime number to appear in the title of a popular song. (The song 867-5309/Jenny peaked at #4 on Billboard in 1982.)

David Radcliffe added “The phone number of Jenny’s twin sister is 8675311” because 8675309 and 8675311 are twin primes.

How likely is it for a telephone number to be prime? The Prime Number Theorem says that for large values of x, the probability that a number less than x is prime is approximately 1/log(x). Since 1/log(107) = 0.062, about 6% of phone numbers are prime.

We could try to be more accurate. We could look at the probability that a seven-digit number is prime rather than simply a number less than 107 (i.e. excluding numbers with less than seven digits). Or we could use the exact number of primes in a certain range (say using Mathematica’s `PrimePi` function) rather than using the Prime Number Theorem approximation. But these refinements would still give us estimates of about 6%. Note that not all seven-digit numbers have been assigned as phone numbers, so an exact calculation still gives only an approximate answer.

What about phone numbers with area codes? The Prime Number Theorem suggests about 4.3% of 10-digit numbers are prime. But the US has on the order of 300 area codes, so most 10-digit numbers are not telephone numbers. Also, area codes were not selected randomly. They were selected with a preference for smaller numbers, which means our estimate of 4.3% may be a little low. (We’d expect more prime numbers to start with small area codes.) But I imagine the area code bias has little effect.