# Do they have a name for these prime numbers?

• Nov 11th 2009, 02:33 AM
Rivelli
Do they have a name for these prime numbers?
I was wondering if there is a name for certain prime numbers such that the sum of the digits of the number equal a prime number.

Example: The prime number 331, 3+3+1=7

What about the added constraint of requiring each digit to be prime as well?

Example: The prime number 23, 2+3 = 5.

Any name for these?
• Nov 11th 2009, 05:45 AM
Media_Man
Looks Fun!
I've never heard of this. Sounds like a fun little relation. Keep in mind, though, that if you generate this list in base 10, you will get a different list than in any other base, making these somewhat less fundamental than perhaps other numbers in number theory. For example, in base 2, \$\displaystyle 7=(111)_2\$, and \$\displaystyle 1+1+1=3\$, so 7 is a Rivelli Number in base 2.

If you can enumerate a list of the first 5 or 10, you can check your list against AT&T's The On-Line Encyclopedia of Integer Sequences If they have never heard of it, they take submissions! You could be the official discoverer of Rivelli Numbers!
• Nov 11th 2009, 03:34 PM
alunw
Not sure about Rivelli's idea, though in a raffle tonight I pointed out that 23 was a nice number because it has prime digits and is 2*prime+1 and prime itself, but I think numbers of the form 111 in some base should have a name, because they are somewhat like Mersenne primes.
Clearly p^n-1 is always divisible by (p-1), but I guess if you take out this factor what you are left with is probably prime under similar conditions to 2^p-1. Similarly a number that is (2*p)^n+1 might be analogous with a Fermat prime.
For instance in base 10 11 is prime, 101 is prime, but I don't know when the next prime of the form 10^n+1 is, or even if there is another one at all.
For Rivelli's idea I guess the ultimate would be to find a big prime number such that the sum of its digits was prime, and the sum of the digits of that was prime and so on. It would certainly be an interesting challenge to come up with numbers you could do this for more than two or three times, though it might well turn out there were plenty of such numbers.