# A Kind of Unique Prime Number

A Prime number with the following properties

Less than 7 digits and more than 3 digits

ALL digits in the number are Prime numbers-- some repeated.

All individual digits in the number add up to a Prime Number-- whose digits also add up to a Prime number

The last 2 digits of the number are same as the first 2 digits.

And, it exhibits another property described by a hyphenated word in Wikipedia.

What is the number and what is that property?

Note: You should be able to get this without the help of computers. But you can do the final check, if you must, on the net once you come up with the answer.

• You say some repeated. That means at least 2 of the numbers add up to an even number. – risky mysteries Jul 23 '20 at 13:02
• So you could have a number ABCADB where A an B are repeated and A+B+C+A+D+B adds up to a Prime number. If that Prime number is say XY then X+Y is also a Prime. Just an example @risk mysteries – DrD Jul 23 '20 at 13:05
• Oh, i thought it meant all sums should be prime :) – risky mysteries Jul 23 '20 at 13:08
• No I meant to add up all the individual digits as above – DrD Jul 23 '20 at 13:10
• It seems that 23 satisfies all the requirements, except maybe the last. But there are plenty of qualifiers that apply, like "non-composite". – Florian F Jul 23 '20 at 13:25

Seems a valid solution is

27527

Less than 7 digits and more than 3 digits

Check

ALL digits in the number are Prime numbers-- some repeated.

Check

All individual digits in the number add up to a Prime Number-- whose digits also add up to a Prime number

23, 5

The last 2 digits of the number are same as the first 2 digits.

Check

And, it exhibits another property described by a hyphenated word in Wikipedia.

It is d-Powerful, because it breaks down as 2^1 + 7^4 + 5^3 + 2^13 + 7^5

As a starting point, the number of digits must be

5 or 6

The first rule tells us that it's

4, 5, or 6. But if it's 4, and the first and last pairs of digits are the same (rule 4), then the sum of the digits is a+b+a+b = 2(a+b) which can't be prime (rule 3).

Given that, the sum of the digits must be

a 2-digit prime, the sum of whose digits is prime. Since the lowest possible number is 22222 and the highest 777777. Though obviously all the digits can't be the same, since then the sum wouldn't be prime.

If it's a 5-digit number

The sum of the digits must be less than 35. The only primes less than 35 whose digit sum is prime are 23 and 29. We're looking for 2(x+y)+z = sum, where all digits are 2 3 5 or 7.

However at that point I'm out of steam because there's multiple combinations there, and even more for 6 digits.

I have no idea about the "hyphenated word" part, which IMO doesn't fit the style of the rest of the problem. What if that wiki page changes its title in the future? What if another "hyphenated word" is added to wiki, ruining the uniqueness?

Without this part, I just ignored the no-computers tag and found the following list (assuming that less than $$7$$ digits means strictly less, and the first and last $$2$$ digits are the same order, i.e. numbers such as $$753257$$ don't count):

27527, 37337, 57557, 572357

Based on the description, I'd guess the intended answer should be

572357, as it has $$6$$ digits.

However a quick google search didn't yield anything for me. Also the number doesn't appear in OEIS.

Thus I put here this result to everyone's reference, which was a comment but has grown too long.

EDIT: Based on the hint given in the comment, I think it is

37337, which is a right-truncatable prime.

Without that hint I would probably never figure out that hyphenated word (and in fact the title of that wiki page is not hyphenated). And of course I cheated in the first place by using a computer :P

• gel erzbivat bar qvtvg sebz nal qverpgvba naq pbagvahr hagvy lbh ner ng gur ynfg qvtvg. What does each number tell you? – DrD Jul 23 '20 at 14:05
• Thanks for the honest comments in the answer. I found this to be a very interesting Prime Number – DrD Jul 23 '20 at 14:20