20230528, 09:55  #89 
"Norman Luhn"
Jan 2007
Germany
769 Posts 
2nd AP27
is known !!!
https://www.primegrid.com/stats_ap26.php 
20230603, 19:31  #90 
"Norman Luhn"
Jan 2007
Germany
769 Posts 
AP23 minimal end is known
It is 403185216600637 + 9523 * 23# * n, n=0..22 and was found first as a normal AP23 in 2006 by Markus Frind.
Joris Perrenet proved this as a minimal on 02 June 2023 ! https://www.pzktupel.de/JensKruseAndersen/RHMINEND.php Last fiddled with by Cybertronic on 20230603 at 19:34 
20230604, 23:47  #91 
Dec 2022
2·3·7·13 Posts 
I assume there is nothing better than exhaustive search to do this; the basic technique is fairly obvious if I don't have it wrong  it just takes (at least) linear time in the size of the number.
One does not say in English 'proved as a minimal'; normal would be 'proved minimal' or 'proved to be minimal', I suppose 'as minimal' or 'as the minimal' would be acceptable or at least better than the version with the indefinite article. An observation I just made is that for all AP longer than 10 for minimal end (6 for minimal start) that record does not also have the minimal difference, and that probably continues forever. But for CPAP, it is likely the exact opposite  the smallest always has minimal difference! 
20230605, 01:11  #92  
Feb 2017
Nowhere
3·13·167 Posts 
Quote:
Here is one way that this could be possible: Considering the APk's for various k, minimal end primes of APk are known for k = 3 to k = 23. The end prime of each such APk may thus properly be referred to as "a minimal" in the sense of being one among many. 

20230605, 04:40  #93 
"Norman Luhn"
Jan 2007
Germany
769 Posts 
@Andrew Usher
I replaced to "proved to be minimal" .... und wehe das stimmt nicht ! Thanks for the hint. 
20230605, 19:40  #94  
Jun 2015
Vallejo, CA/.
2^{3}·3·7^{2} Posts 
Quote:


20230610, 01:04  #95 
Dec 2022
1042_{8} Posts 
I know that English is not his first language, and that's why I think correcting him may be helpful and not just rude (Actually, I'd correct a native speaker, too, if one wrote those exact words, but that would not happen.). However, it's more interesting that I seem to have discovered a new rule (or tendency) of English grammar, on something I'd never thought about until his phrase struck my ear as absolutely impossible.

20230610, 05:06  #96 
"Norman Luhn"
Jan 2007
Germany
769 Posts 
Guys, but am rather pleased when suggestions for a better wording come.
BTW, "proved this as a minimal" was a onlinetranslater suggestion. In German, we have hundreds of ways to formulate an issue. The translators then have to make something out of it. Last fiddled with by Cybertronic on 20230610 at 05:08 
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