Prime 'anagrams'

[raresaturn]

Take a prime number and rearrange the digits to get another prime number. This is difficult for small numbers, but gets easier as the primes get larger. Therefore the question becomes: What is that largest prime that has no anagrams? ie. no other primes can be made by reordering it's digits. It might seem trivial for a number like 22222222222222221 to make the last digit even, but is such a number prime in the first place? (I haven't checked this example LOL)

I wrote a little Python script to check these, so far the largest I've found with zero anagrams is 33343 (it's a slow program )

Can anyone find larger ones?

[axn]

99949999 appears to be the largest 8-digit one

[raresaturn]

Originally Posted by axn

99949999 appears to be the largest 8-digit one

Cool, how did you find that one?

[axn]

List of near-repdigit primes/PRPs (https://stdkmd.net/nrr/prime/primesize.txt) might be a good place to look

[axn]

Originally Posted by raresaturn

Cool, how did you find that one?

Not by looking at all the anagrams for a prime, for sure

I looped thru all primes < 10^8, converted them into a "canonical" form and checked if that has been seen before. Any canonical form seen only once means, it has no anagrams. Of course, you get to know that only after entire range of n-digit primes have been scanned.

[raresaturn]

Originally Posted by axn

Not by looking at all the anagrams for a prime, for sure

I de-duped my lists, so having all the same digits doesn't really count (or at least not in the spirit of the task )
EDIT: i think i responded the the wrong msg, no matter

[ATH]

As you pointed out any prime with all even numbers or 5's and just the last digit 1,3,7 or 9 are trivial candidates for this.

Non-trivial ones:
Largest below 10⁶: 999499
Largest below 10⁷: 9999991
Largest below 10⁸: 99949999
Largest below 10⁹: 999499999

There are "only" 350 of them from 11 to 10⁹ including the trivial ones.
Count of them including trivials starting from 11:
10¹ - 10²: 13
10² - 10³: 34
10³ - 10⁴: 45
10⁴ - 10⁵: 68
10⁵ - 10⁶: 67
10⁶ - 10⁷: 47
10⁷ - 10⁸: 36
10⁸ - 10⁹: 40

[raresaturn]

Originally Posted by ATH

As you pointed out any prime with all even numbers or 5's and just the last digit 1,3,7 or 9 are trivial candidates for this.

Non-trivial ones:
Largest below 10⁶: 999499
Largest below 10⁷: 9999991
Largest below 10⁸: 99949999
Largest below 10⁹: 999499999

That's really interesting...I wonder if we can just keep adding 9's to the end eg: 99949999999999999999999999999 (and will it always be a prime?)

[LaurV]

Nope.

[kriesel]

Originally Posted by raresaturn

It might seem trivial for a number like 22222222222222221 to make the last digit even, but is such a number prime in the first place? (I haven't checked this example LOL)

22222 222222 222221 = 3 × 23 × 211 × 239851 × 6 363769 per https://www.alpertron.com.ar/ECM.HTM
Trimming away a 2 at a time, I didn't encounter a prime until 2221.

[science_man_88]

forperm(digits(randomprime(10^9)),x,print(x))