# Thread: Special Prime Numbers

1. ## Special Prime Numbers

Special prime numbers
Hello to everyone.
I'm studying a series of special prime numbers and I managed to make a conjecture by experiment that unfortunately did not manage to prove or to disprove it.

Let me explain what it is.

Take any prime number such as 59, and then put to his right alongside a number between 1 3 7 9 so the new number is still prime, for example 593, and iterative reasoning until we get no prime number. In our case 59393339 because 593933391, 593933393, 593933397 and 595933399 are not primes.
We say that the prime number 59393339 was generated by using this algorithm from the prime 59 and 59 is the generator 59393339 always through the algorithm.
My guess says this: Whatever is the starting prime number, the algorithm always arrives at the end.

We do note that there are some prime numbers that are generated by themselves through the algorithm, which means that the algorithm stops immediately at the first step. For example: 89, because 891,893,897,899 are not prime. It's demostrated that there are infinite prime numbers of this type.

Thank you

Picozzi

2. It's extremely likely that your guess is correct, although it's probably a big hassle to prove.

Note that your starting prime, if it's different from 3, is always $\displaystyle \equiv \pm 1 \mod 3$; since an integer is congruent (mod 3) to the sum of its decimal digits, that leaves you the possibility of using the digits {1,7} at most once at some later step in your algorithm : never if your starting prime is congruent to -1 (mod 3), and at most once if it's congruent to 1 (mod 3). For example $\displaystyle 59 \equiv -1 \mod 3$ so you can only use the digits {3,9}.

Note also that conjectures dependent on the decimal base are considered relatively uninteresting because the decimal base is arbitrary.
And in all cases, to quote Hardy in his book Ramanujan (do not take offence!):

It is comparatively easy to make clever guesses; indeed there are theorems, like "Goldbach's Theorem", which have never been proved and which any fool could have guessed.

3. OK, thank you for the answer. But I have another funny question:

Can I become famous for this simply conjecture, like Goldbach?

4. I'm not sure if you're serious, but in my opinion this conjecture will not make you famous.

5. No, I was only joking.

However, I tryed the algorithm in other bases, such as in binary (we can only put 1 on right sight), and it still works. it always ends.

6. As I said, it's not surprising.

7. Originally Posted by Bruno J.
It is comparatively easy to make clever guesses; indeed there are theorems, like "Goldbach's Theorem", which have never been proved and which any fool could have guessed.
It hasn't been proved (yet), so it's not a theorem, but still a conjecture...