- --- On Sun, 5/15/11, Kermit Rose <kermit@...> wrote:
> > Centuries refer to ranges [100X+1,100X+99].

Imagine a fingerprint of "primes in a century". That's it. Nothing more

> > Does anyone happen to know where the first repetition of the primes

> > in a century occurs? I see no reason to suspect that this would

> > necessarily happen to be from the first 2 void centuries. A very

> > minor point in the prime universe perhaps, but one I have long

> > wondered about.

>

> I do not understand the question.

> What do you mean by the first 2 void centuries.

100x to 100x+99 all being void of primes, obviously.

> In the range [1,99] , 3 is prime,

Yeah, but 2 is prime, and 102 is composite. And 5 is prime but 105 is composite. Clearly the [0..100) century is not the same is the [100-200) century.

> and in the range [101,199], 103 is prime.

Obviously, 2 consecutive centuries completely void of primes would have an identical fingerprint ("no primes"), the question is whether there's anything before that which satisfies the 100-way equivalence.

Phil > From: Jens Kruse Andersen <jens.k.a@...>

Being a repetition of?

> > Centuries refer to ranges [100X+1,100X+99]. Does anyone happen to know

> > where the first repetition of the primes in a century occurs? I see no reason to

> suspect that this would necessarily happen to be from the first 2 void centuries.

> > A very minor point in the prime universe perhaps, but one I have long wondered about.

>

> There is no case of consecutive centuries with more than 5

> primes below 10^12.

>

> The following assumes that non-consecutive centuries are

> allowed.

>

> The first case measured by the larger century:

> {390500, 480800} + {3, 27, 39, 53, 81}

> The first case measured by the average:

Average of what? (Average seems a strange metric to use in discrete mathematics)

> {78900, 578700} + {1, 19, 29, 41, 77, 79, 89}

> The first repetition of the first century with less than 16

So by "repetition" do you mean that that's a patten in consecutive centuries? Given that you jumped on his wording earlier you could at least be explicit yourself.

> primes

> (meaning I could quickly find a repetition):

> {500, 47843760324362600} + {3, 9, 21, 23, 41, 47, 57, 63,

> 69, 71, 77, 87, 93, 99}

Phil- --- On Sun, 5/15/11, Jens Kruse Andersen <jens.k.a@...> wrote:
> > Centuries refer to ranges [100X+1,100X+99]. Does anyone happen to know

Why doesn't {72676000, 72676100} + {33, 57, 69, 81} appear between those two? And {177343900, 177344000} + {9, 39, 93} also?

> > where the first repetition of the primes in a century occurs? I see no reason to

> > suspect that this would necessarily happen to be from the first 2 void centuries.

> > A very minor point in the prime universe perhaps, but one I have long wondered about.

>

> Assuming you mean two consecutive centuries, the first

> cases are:

> {47326700, 47326800} + {}

> {72676000, 72676100} + {33, 57, 69, 81}

> {177343900, 177344000} + {9, 39, 93}

> {180882800, 180882900} + {17}

> {191912800, 191912900} + {}

>

> First case with 0 to 5 primes in the century:

> 0: {47326700, 47326800} + {}

> 1: {180882800, 180882900} + {17}

Phil - Phil Carmody wrote:
>> From: Jens Kruse Andersen <jens.k.a@...>

480800 + {3, 27, 39, 53, 81} being a repetition of 390500 + {3, 27, 39, 53, 81}.

>> > Centuries refer to ranges [100X+1,100X+99]. Does anyone happen to know

>> > where the first repetition of the primes in a century occurs? I see no reason to

>> suspect that this would necessarily happen to be from the first 2 void centuries.

>> > A very minor point in the prime universe perhaps, but one I have long wondered

>> > about.

>>

>> The following assumes that non-consecutive centuries are

>> allowed.

>>

>> The first case measured by the larger century:

>> {390500, 480800} + {3, 27, 39, 53, 81}

>

> Being a repetition of?

As you quoted me saying:>> The following assumes that non-consecutive centuries are allowed.

This turned out to be the intended interpretation when the original poster

later gave a decade example saying:> The first pattern repetition is {53,59} following {23,29}

My former case said "measured by the larger century", referring

>> The first case measured by the average:

>> {78900, 578700} + {1, 19, 29, 41, 77, 79, 89}

>

> Average of what? (Average seems a strange metric to use in discrete mathematics)

to 480800 in {390500, 480800}.

Thereafter "measured by the average" meant "measured by the

average of the two centuries":

(78900+578700)/2 < (390500+480800)/2

>> The first repetition of the first century with less than 16

This is still following my "The following assumes that non-consecutive

>> primes

>> (meaning I could quickly find a repetition):

>> {500, 47843760324362600} + {3, 9, 21, 23, 41, 47, 57, 63,

>> 69, 71, 77, 87, 93, 99}

>

> So by "repetition" do you mean that that's a patten in consecutive centuries? Given

> that you jumped on his wording earlier you could at least be explicit yourself.

centuries are allowed."

47843760324362600 + {3, 9, 21, 23, 41, 47, 57, 63, 69, 71, 77, 87, 93, 99}

is a repetition of

500 + {3, 9, 21, 23, 41, 47, 57, 63, 69, 71, 77, 87, 93, 99}

It was an added curio about a small starting century (instead of the

smallest ending or smallest average), but it doesn't answer any

interpretation of the request for "the first repetition".

The century starting at 0 is obviously inadmissible. The century starting

at 100 is the first admissible but it has 21 primes and that would be

computationally extremely hard to repeat although the k-tuple

conjecture predicts infinitely many cases.

>> First case with 0 to 5 primes in the century:

My post continued:

>> 0: {47326700, 47326800} + {}

>> 1: {180882800, 180882900} + {17}

> Why doesn't {72676000, 72676100} + {33, 57, 69, 81} appear

> between those two? And {177343900, 177344000} + {9, 39, 93} also?

>> 2: {251848800, 251848900} + {1, 43}

It was sorted by the number of primes "0 to 5" and gave the first case

>> 3: {177343900, 177344000} + {9, 39, 93}

>> 4: {72676000, 72676100} + {33, 57, 69, 81}

>> 5: {3451361900, 3451362000} + {11, 17, 23, 59, 71}

with that number of primes.

--

Jens Kruse Andersen --- On Mon, 5/16/11, Jens Kruse Andersen <jens.k.a@...> wrote:

[SNIP - me showing the mental faculties of your average seafood]

> The century starting at 0 is obviously inadmissible. The

> century starting

> at 100 is the first admissible but it has 21 primes and

> that would be

> computationally extremely hard to repeat although the

> k-tuple

> conjecture predicts infinitely many cases.

>

> >> First case with 0 to 5 primes in the century:

> >> 0: {47326700, 47326800} + {}

> >> 1: {180882800, 180882900} + {17}

>

> > Why doesn't {72676000, 72676100} + {33, 57, 69, 81} appear

> > between those two? And {177343900, 177344000} + {9, 39, 93} also?

>

> My post continued:

> >> 2: {251848800, 251848900} + {1, 43}

> >> 3: {177343900, 177344000} + {9, 39, 93}

> >> 4: {72676000, 72676100} + {33, 57, 69, 81}

> >> 5: {3451361900, 3451362000} + {11, 17, 23, 59, 71}

>

> It was sorted by the number of primes "0 to 5" and gave the

> first case with that number of primes.

Again, extreme short-sightedness, I just presumed they'd be in numerical order.

I'm pretty sure I confused the "send" and "cancel" buttons, as I'm sure I had shed most my confusion at some point, and thus my silly questions were unnecessary.

It appears you've only attacked this from an angle of finding the easiest pattern to detect, rather than an exhaustive one. As primes tend towards sparseness, this is clearly the optimal approach. However, with that view-point, the denser patterns are the more interesting ones - I wonder what remains yet undiscovered?

Sorry for being dense. In my defence, the score was 6-1, so it was a very long night.

Phil