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I thought up of a sequence of 6 primes which are the following:

2, 3, 5, 7, 37, and 73.

These 6 primes are the primes where you can combine their digits in any way, including by removing digits (but not including removing all of the digits), and you will still be left with a prime. I also created a proof that proves these are the only 6 primes with the property.

But my question is what the title is: is there a name for these primes?

 

If you are wondering, this is the proof:

Primes are numbers whose only factors are 1 and itself. Single digit primes have only one digit, so there is only one way to combine its digits and that combination is itself, so all single digit primes are in the group.

The slightly complicated part is for 2 or more digit primes. If a prime has 2 or more digits, it will have more than one combination. Since you can only have 1, 3, 7, and 9 as the last digits, your options are limited. Removing digits for a combination is allowed, so the last digit has to also be prime. Of those four, only 3 and 7 are primes, so the number has to end in 3 or 7. Although, more valid combinations are if any digit is transferred to the back of the number to create a valid combination and any of those digits can be the only digit left. This restricts what all digits can be to 3 and 7. Now, 33 and 77 aren't prime since they are multiples of 11. This leaves only 2 options out of all 2 digit numbers that are eligible to be in this list, and they are 37 and 73.

Finally, there aren't any higher digit numbers that can be qualified for this list because all digits of 2+ digit numbers have to be 3 or 7, but this means there has to be at least 2 or more digits with the number 3 or the number 7 in them. The reason is because a valid combination is removing all but 2 digits of the number, and those 2 digits left can be any 2 of the digits present in the number. Let's use 337. There are 6 combinations that are valid by deleting one digit and arranging the rest, which are the following: 33, 37, 37, 73, 73, 33. But there is a problem: 33 is not prime, and neither is 77. Since all 3+ digit numbers have to have only 3 or 7 as all their digits, there HAS to be 2 or more digits with 3 or 7, but that means at least one of their combinations won't be prime and that makes no 3+ digit prime eligible for this list.

2: 2

3: 3

5: 5

7: 7

37: 37, 73, 3, 7

73: 73, 37, 7, 3

creepercraft97T3  Sep 18, 2018
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This large number(24 digits) is a prime number and is called, as you know, a "Truncatable Prime":

357,686,312,646,216,567,629,137

It has a very special and unique property which is this: If you start from the left and remove the first 3, the remaining number is STILL a prime number! If you remove the next number 5, the remaining number is still a prime number! If you continue and remove the next number 7, the remaining number is still a prime number. And if you continue with this process, the remaining number continues to be a prime number, all the way to the last digit of 7!!!. 

Isn't that remarkable?? Your sequence of primes is something like that, but much less impressive!!!.

Guest Sep 18, 2018
edited by Guest  Sep 18, 2018

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