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2022-11-20, 19:52   #56
raresaturn

Jul 2021

2416 Posts

Quote:
 Originally Posted by MattcAnderson (2*10^n - 11)/9 is a prime number for n=4, 18, 100, 121, ... so the number 2221 is a prime number as well as others. Cheers
As Batalov mentioned I don' think you understand the thread. You have to re-arrange the digits to make a prime anagram. Using even numbers is pointless in this regard. See first post in this thread for more info.

2022-11-21, 00:39   #57
SmartMersenne

Sep 2017

13910 Posts

Quote:
Originally Posted by raresaturn
Quote:
 Originally Posted by MattcAnderson (2*10^n - 11)/9 is a prime number for n=4, 18, 100, 121, ... so the number 2221 is a prime number as well as others. Cheers
As Batalov mentioned I don' think you understand the thread. You have to re-arrange the digits to make a prime anagram. Using even numbers is pointless in this regard. See first post in this thread for more info.
I think MattcAnderson is trying to guarantee no prime anagrams, as all other anagrams are even.

2022-11-21, 02:32   #58
raresaturn

Jul 2021

22·32 Posts

Quote:
 Originally Posted by SmartMersenne I think MattcAnderson is trying to guarantee no prime anagrams, as all other anagrams are even.
yes but using even numbers is trivial.. why bother?

2022-11-21, 04:02   #59
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

3·3,329 Posts

Quote:
 Originally Posted by raresaturn yes but using even numbers is trivial.. why bother?
You still have to find a prime then it is trivial.
Have you?

2022-11-21, 04:20   #60
raresaturn

Jul 2021

22×32 Posts

Quote:
 Originally Posted by Batalov You still have to find a prime then it is trivial. Have you?
Yes. All numbers ending in 2 are not prime. What's the point if rearranging the prime just gives you something like 122222222 ????. There is no possibility of a prime if the last digit is even.

2022-11-21, 04:51   #61
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

3·3,329 Posts

but (2*10^282948-11)/9 i.e. 2....{282947 of them}...21 is prime. What does it tell you?

Quote:
 Originally Posted by raresaturn Take a prime number and rearrange ... Therefore the question becomes: What is that largest prime that has no anagrams? Can anyone find larger ones?
Who wrote that?

2022-11-21, 05:32   #62
raresaturn

Jul 2021

22·32 Posts

Quote:
 Originally Posted by Batalov but (2*10^282948-11)/9 i.e. 2....{282947 of them}...21 is prime. What does it tell you?
That there's no need to check for anagrams because any rearrangement will make the number even.

Quote:
 Originally Posted by Batalov Who wrote that?
Apart from trivial ones, obviously.

Last fiddled with by raresaturn on 2022-11-21 at 05:35

 2022-11-21, 08:18 #63 LaurV Romulan Interpreter     "name field" Jun 2011 Thailand 3×5×683 Posts I bet I know a prime at least as large as any prime you know, or larger, which has no anagrams. But there is a catch... It is in base 2...
2022-11-21, 16:02   #64
Dr Sardonicus

Feb 2017
Nowhere

10111110011102 Posts

Quote:
 Originally Posted by LaurV I bet I know a prime at least as large as any prime you know, or larger, which has no anagrams. But there is a catch... It is in base 2...
Good one!

For n > 1, there is ("Bertrand's Postulate") at least one prime between 2^(n-1) and 2^n. These are the primes with precisely n bits. My rough-and-ready guess is that, like the integers in (2^(n-1),2^n), the number of binary 1's in these primes is highly concentrated around n/2.

An actual count for n = 27 gives the following totals for numbers of 1's from 1 to 27.

[0, 0, 2, 40, 279, 1291, 5706, 20287, 51822, 117870, 224567, 355931, 487555, 563833, 558175, 489259, 355960, 215725, 119409, 51953, 18651, 5720, 1498, 178, 33, 0, 0]

There are no odd primes with exactly one binary one. The only possible primes with two binary ones are the Fermat primes. There is a decent chance of a single n-bit prime with exactly three binary ones (The first and last bit have to be 1, leaving one 1 among the remaining bits.)

The only primes with no binary digits of 0 are the Mersenne primes.

There is exactly one 25-bit prime p with 3 binary ones. In decimal, with vector of binary digits,

16777729 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1]

And there is at least a chance that there is precisely one n-bit prime with exactly n-1 binary ones.

This is realized, again with n = 25:

33546239 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

 2022-11-25, 19:28 #65 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3·3,329 Posts Here's the apparent champion of primes* for which all anagrams are composite. It was found by unconnected a few years ago: 68400000000000000000000...0000000000000000001 (1'127'121 decimal digits, and notably five distinct digits are used). I think this is a rather good closing point. *and it is a proven prime, too, unlike other near/quasi-repunits which are PRPs
 2022-11-25, 19:53 #66 kruoli     "Oliver" Sep 2017 Porta Westfalica, DE 4C516 Posts If I put all the zeros in front and put the remaining digits in order I get 00...0006841, which is also prime. Maybe others numbers, too, where we do not need to start with zero as a digit. As long as we have the 1 at the end, why shouldn't there be other prime possible? Have you checked all others? Last fiddled with by kruoli on 2022-11-25 at 19:55 Reason: Grammar.

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