mersenneforum.org Primes in centuries: The Tortoise and the Hare
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2022-01-05, 18:41   #12
mart_r

Dec 2008
you know...around...

13448 Posts

Quote:
 Originally Posted by kog67 I have found the 3 missing terms :
Excellent work! Thanks
That brings the list I started back in around 2006 to completion

2022-01-09, 13:22   #13
LaurV
Romulan Interpreter

"name field"
Jun 2011
Thailand

233518 Posts

Quote:
 Originally Posted by mart_r Who doesn't?
Well, my (joke) answer was more along the lines "100 factorial", but thanks for the cycles anyhow (to you and the others)
Trivia: (hihi) Dr.S sent me by PM, long before you posted, the same century you found. I honestly didn't imagine the answer would be so low (actually, didn't think much about). I replied him the same like here.

2022-01-09, 14:40   #14
mart_r

Dec 2008
you know...around...

22×5×37 Posts

Quote:
 Originally Posted by LaurV Well, my (joke) answer was more along the lines "100 factorial", but thanks for the cycles anyhow (to you and the others)
No cycles involved (well, not recently), I've known that first century without primes by heart for more than 20 years now
Actually, if you weren't sure whether 100!+/-1 were prime or not, you'd have to go for 199!, or 199# (or larger, of course).

2022-01-09, 15:49   #15
Dr Sardonicus

Feb 2017
Nowhere

3×17×113 Posts

Quote:
 Originally Posted by mart_r Actually, if you weren't sure whether 100!+/-1 were prime or not, you'd have to go for 199!, or 199# (or larger, of course).
I am certain that 100! + 1 is not prime. No cycles required.

Clearly 100! + 1 > 101, and is divisible by 101, by Wilson's Theorem.

2022-01-09, 20:12   #16
mart_r

Dec 2008
you know...around...

13448 Posts

Quote:
 Originally Posted by Dr Sardonicus I am certain that 100! + 1 is not prime. No cycles required. Clearly 100! + 1 > 101, and is divisible by 101, by Wilson's Theorem.
Sorry, hadn't taken Wilson's Theorem into account.

Quote:
 Originally Posted by Dr Sardonicus The following example gives a 21-tuplet of which 20 lie between consecutive multiples of 100. 39433867730216371575457664399 + [0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84]
That one also comes close, with 20 primes just inside the "century":
622803914376064301858782434517 + [0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76, 82, 84]
BTW, the first one with 19 primes is <= 110885131130067570042700 + [3, 7, 9, 13, 19, 21, 27, 31, 33, 37, 43, 49, 51, 57, 61, 63, 69, 91, 97], see A186311.

2022-01-14, 00:11   #17
Bobby Jacobs

May 2018

1000000102 Posts

Quote:
 Originally Posted by mart_r Who doesn't? Code: (20:40) gp > primepi(1671900)-primepi(1671800) %1 = 0
A lot of people never know about a century with no primes.

 2022-01-14, 06:22 #18 LaurV Romulan Interpreter     "name field" Jun 2011 Thailand 100110111010012 Posts Well, talking about 100 years with no primes (i.e. no round century), I think the first gap >= 100 is at 370262. (in table at position 17) Quick pari code: Code: p=2; forprime(q=3,10^8,if(q-p>=100,print(p);break);p=q) Last fiddled with by LaurV on 2022-01-14 at 06:26
 2022-01-24, 18:27 #19 Bobby Jacobs     May 2018 1000000102 Posts If it is not required to be between multiples of 100, then there are some interesting spans of 100 numbers with a lot of primes. For example, there are a whopping 19 primes from 2657 to 2753.

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