## Re: [PrimeNumbers] First repetition of prime pattern within "centuries"

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• ... I see only the void centuries as repetitions. I try to build a counterexample in my head but fail to do so, at least for consecutive centuries. Did I
Message 1 of 11 , May 15, 2011
On Sun, 15 May 2011 09:54:34 -0000, woodhodgson@... wrote:
> 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.

I see only the void centuries as repetitions.
I try to build a counterexample in my head but fail to do so,
at least for consecutive centuries. Did I understand the question
incorrectly ?
Repetitions would make the numbers of the interval composite.
I'm too tired to dig further; could anybody else help ?

yg
• ... Assuming you mean two consecutive centuries, the first cases are: {47326700, 47326800} + {} {72676000, 72676100} + {33, 57, 69, 81} {177343900, 177344000}
Message 2 of 11 , May 15, 2011
woodhodgson wrote:
> 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.

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}
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}

There is no case with more than 5 below 10^11. There are 5 cases with 5:
{3451361900, 3451362000} + {11, 17, 23, 59, 71}
{34221969900, 34221970000} + {7, 19, 31, 49, 73}
{41290268400, 41290268500} + {1, 7, 19, 31, 73}
{54757509100, 54757509200} + {27, 39, 81, 93, 99}
{94596013600, 94596013700} + {9, 39, 63, 69, 93}

Three consecutive centuries is only possible with no primes since 3
always divides one of {n, n+100, n+200}.

--
Jens Kruse Andersen
• ... 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
Message 3 of 11 , May 15, 2011
woodhodgson wrote:
> 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:
{78900, 578700} + {1, 19, 29, 41, 77, 79, 89}

The first repetition of the first century with less than 16 primes
(meaning I could quickly find a repetition):
{500, 47843760324362600} + {3, 9, 21, 23, 41, 47, 57, 63, 69, 71, 77, 87, 93, 99}

--
Jens Kruse Andersen
• woodhodgson@xtra.co.nz rupert.weather@gmail.com woodhodgson@xtra.co.nz Date: Sun May 15, 2011 2:54 am ((PDT)) ... I do not understand the question. What do
Message 4 of 11 , May 15, 2011
"woodhodgson@..." rupert.weather@... woodhodgson@...
Date: Sun May 15, 2011 2:54 am ((PDT))
says:

> 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

I do not understand the question.

What do you mean by the first 2 void centuries.

In the range [1,99] , 3 is prime,
and in the range [101,199], 103 is prime.

Could you explain in more concrete terms exactly your question?

Kermit Rose
• A more rigorous definition is: Numbers x such that for all N in [100x,100x+99], N is prime iff N+100 is prime. This sequence contains in particular the first
Message 5 of 11 , May 15, 2011
A more rigorous definition is:

Numbers x such that for all N in [100x,100x+99], N is prime iff N+100 is prime.

This sequence contains in particular the first of two consecutive prime-free centuries, i.e., N such that there is no prime in [100 N, 100 (N+2)], cf. oeis.org/A181098 .

M.

--- In primenumbers@yahoogroups.com, Kermit Rose <kermit@...> wrote:
>
>
> "woodhodgson@..." rupert.weather@... woodhodgson@...
> Date: Sun May 15, 2011 2:54 am ((PDT))
> says:
>
> > 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
>
> I do not understand the question.
>
> What do you mean by the first 2 void centuries.
>
> In the range [1,99] , 3 is prime,
> and in the range [101,199], 103 is prime.
>
>
>
> Could you explain in more concrete terms exactly your question?
>
> Kermit Rose
>
• ... Imagine a fingerprint of primes in a century . That s it. Nothing more ... 100x to 100x+99 all being void of primes, obviously. ... Yeah, but 2 is prime,
Message 6 of 11 , May 15, 2011
--- On Sun, 5/15/11, Kermit Rose <kermit@...> wrote:
> > 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
>
> I do not understand the question.

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

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

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
• ... Being a repetition of? ... Average of what? (Average seems a strange metric to use in discrete mathematics) ... So by repetition do you mean that that s
Message 7 of 11 , May 15, 2011
> From: Jens Kruse Andersen <jens.k.a@...>
> > 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}

Being a repetition of?

> 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)

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

Phil
• ... Why doesn t {72676000, 72676100} + {33, 57, 69, 81} appear between those two? And {177343900, 177344000} + {9, 39, 93} also? Phil
Message 8 of 11 , May 15, 2011
--- 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
> > 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}

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

Phil
• ... 480800 + {3, 27, 39, 53, 81} being a repetition of 390500 + {3, 27, 39, 53, 81}. ... This turned out to be the intended interpretation when the original
Message 9 of 11 , May 15, 2011
Phil Carmody wrote:
>> From: Jens Kruse Andersen <jens.k.a@...>
>> > 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
>>
>> 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?

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

>> 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)

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

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

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:
>> 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.

--
Jens Kruse Andersen
• ... [SNIP - me showing the mental faculties of your average seafood] ... Again, extreme short-sightedness, I just presumed they d be in numerical order. I m
Message 10 of 11 , May 16, 2011
--- 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
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