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Old 2022-11-20, 19:52   #56
raresaturn
 
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Quote:
Originally Posted by MattcAnderson View Post
(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.
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Old 2022-11-21, 00:39   #57
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Quote:
Originally Posted by raresaturn View Post
Quote:
Originally Posted by MattcAnderson View Post
(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.
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Old 2022-11-21, 02:32   #58
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Quote:
Originally Posted by SmartMersenne View Post
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?
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Old 2022-11-21, 04:02   #59
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Quote:
Originally Posted by raresaturn View Post
yes but using even numbers is trivial.. why bother?
You still have to find a prime then it is trivial.
Have you?
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Old 2022-11-21, 04:20   #60
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Quote:
Originally Posted by Batalov View Post
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.
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Old 2022-11-21, 04:51   #61
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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 View Post
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?
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Old 2022-11-21, 05:32   #62
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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 View Post
Who wrote that?
Apart from trivial ones, obviously.

Last fiddled with by raresaturn on 2022-11-21 at 05:35
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Old 2022-11-21, 08:18   #63
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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...
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Old 2022-11-21, 16:02   #64
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Quote:
Originally Posted by LaurV View Post
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]
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Old 2022-11-25, 19:28   #65
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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
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Old 2022-11-25, 19:53   #66
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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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