Primes in centuries: The Tortoise and the Hare

[mart_r]

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

[LaurV]

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.

[mart_r]

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).

[Dr Sardonicus]

Originally Posted by mart_r

<snip>
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.

[mart_r]

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.

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.

[Bobby Jacobs]

Originally Posted by mart_r

Who doesn't?

(20:40) gp > primepi(1671900)-primepi(1671800)
%1 = 0

A lot of people never know about a century with no primes.

[LaurV]

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:

p=2; forprime(q=3,10^8,if(q-p>=100,print(p);break);p=q)

[Bobby Jacobs]

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.