# Prime Number with Prime Digits

• fibonacci235
In summary, the speakers discuss whether there is a specific name for prime numbers whose digits are all prime. They also mention other types of primes and their names, such as Mersenne primes and palindromic primes. The concept of "Happy numbers" is brought up and one speaker suggests naming these types of primes "Digital Primes" in base 10. The conversation also touches on the idea that just because a number has a name does not necessarily mean it is worth studying.
fibonacci235
Is there a name for a prime number whose digits are all prime? The first several that I can think of are

2,3,5,7 and 23, 23 being the first double digit prime whose digits are all prime.

Well, I don't think so.
Reason?
It ought to be just an artefact of the particular number representation system you use.

If you let "11" be your base number, rather than 10, your "23" will be written as 21, the latter digit not being prime at all.

The quality of a number being prime is independent of its particular representation, but you can't expext the same for the digits of two distinct representations of the same number.

I know that a prime number doesn't depend on its representation, but number theorists have names for different types of primes i.e. Mersenne primes, Irregular primes, Safe primes, and my personal favorite, Sexy primes. I just wanted to know if there was a name for this particular type of prime. For example, as you probably well know there is a name for numbers like 222. They're called palindromic numbers. Now, 313 is a palindromic number but it is also prime. I know that 313 is prime independently of being a palindromic number. That's just a coincidence. My point is, is there a specific name for primes whose digits are all prime? That's all I'm getting at. This also raises the question, are there an infinite number of palindromic primes? Maybe, for the sake of precision, I should have posed my question this way: In a base 10 number system is there a name for a prime number whose digits are all prime?

Mersenne primes are Mersenne primes in all number systems, and so on.

Sure, some of dubbed palindromic numbers as being palindromic. But, just because you set a name on something doesn't mean it constitutes anything worthwhile studying (i.e, nobody studies palindromic numbers)

The reason why palindromic numbers have gotten their name is that the property of palindromy is well known outside maths, so it sort of stuck.

I would be extremely surprised if anyone has given your types of perfectly definable numbers a particular name (I'm not at all saying your question was vague or anything. It wasn't, your first post was perfectly clear)

If you count 1 as not being prime, then numbers of this type can't exist in binary. The larger your base is, the more such numbers can exist in that base, because you have more single digit primes. Example, in Hex, B and D are prime. (Funny, I always thought of them as being "even letters".)

This sounds like "Happy numbers" which are also base dependent.

arildno said:
Mersenne primes are Mersenne primes in all number systems, and so on.

Sure, some of dubbed palindromic numbers as being palindromic. But, just because you set a name on something doesn't mean it constitutes anything worthwhile studying (i.e, nobody studies palindromic numbers)

The reason why palindromic numbers have gotten their name is that the property of palindromy is well known outside maths, so it sort of stuck.

I would be extremely surprised if anyone has given your types of perfectly definable numbers a particular name (I'm not at all saying your question was vague or anything. It wasn't, your first post was perfectly clear)

Then I think I will name them Digital Primes.

## 1. What is a prime number with prime digits?

A prime number with prime digits is a positive integer that is only divisible by 1 and itself, and all of its digits are also prime numbers (2, 3, 5, or 7).

## 2. How many prime numbers with prime digits are there?

There are 23 prime numbers with prime digits between 0 and 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, and 83.

## 3. Are all prime numbers with prime digits also palindromes?

No, not all prime numbers with prime digits are palindromes. A palindrome is a number that reads the same backward and forward. Only 2, 3, 5, and 7 are palindromes among the prime numbers with prime digits.

## 4. Can a prime number with prime digits have more than 2 digits?

Yes, a prime number with prime digits can have more than 2 digits. For example, 23, 29, and 73 are all prime numbers with prime digits.

## 5. How are prime numbers with prime digits important in mathematics?

Prime numbers with prime digits have applications in cryptography, which is used to secure data and communications. They are also used in generating random numbers, which is important for simulations and statistical analysis. Additionally, the study of prime numbers with prime digits helps to deepen our understanding of number theory and patterns in mathematics.

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