Counting up in prime numbers #2

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Re: Counting up in prime numbers #2

Post by BlackTuesday »

RogerE wrote:
15 Jul 2020 12:25
Recall that two primes in a run of three integers (>2) are twin primes.
.
.
Thanks RogerE! :D

And guess what, twin primes again :D

114,641

114,643

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Re: Counting up in prime numbers #2

Post by RogerE »

Excellent! This neighbourhood is quite crowded...

114,649

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Re: Counting up in prime numbers #2

Post by RogerE »

After almost 3.5 days:

TWIN PRIMES!

114,659

114,661

/RogerE :D

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Re: Counting up in prime numbers #2

Post by BlackTuesday »

114,671

:)

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Re: Counting up in prime numbers #2

Post by RogerE »

114,679

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Re: Counting up in prime numbers #2

Post by BlackTuesday »

Twin primes again :D

114,689

114,691

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Re: Counting up in prime numbers #2

Post by RogerE »

Wow! That last "century" was very prime-rich!
Before we move on, let's review that "century".
The recent posts found that there are 13 primes in the interval [114599, 114699],
of which 8 occur as twin primes, 2 as "near twins" and just 3 occur as "isolates".
That is a strikingly "dense" constellation of double stars.

Here is a "heuristic" view of what is going on.
Suppose the number N is divisible by several small primes, and the number d is a
small number not divisible by any of those primes. Then N – d and N + d have a
"good chance" of being primes, because they cannot be divisible by any of the prime
factors of N or d
.
Suppose q is the smallest prime which is not a factor of N. If d is a small number between
2 and q–squared which is not divisible by any prime factor of N, then d must itself be prime.
In particular, if p is a small prime which is not a factor of N, then the numbers N – p and N + p
have a "good chance" of being primes.
(Of course, d = 1 also "works", though we conventionally do not call 1 a prime.)

Let's apply this reasoning to the case N = 2x2x3x3x5x7x7x13, so N = 114660 and q = 11.
We expect the numbers 114660 – d and 114660 + d to have a "good chance" of being prime
when d = 1 or d = p is any prime which is not a factor of N and is less that 11^2 = 121,
that is, p = 11 or 17 ≤ p ≤ 113.
It turns out that the following instances are primes:
N – 1 = 114659, N + 1 = 114661 (twins)
N – 11 = 114649 (isolate)
N - 17 = 114643, N – 19 = 114641 (twins)
N – 43 = 114617, N – 47 = 114613 (near twins)
N – 59 = 114601, N – 61 = 114599 (twins)
The earlier "century" contains isolated primes at
N – 67 = 114593, N – 83 = 114577, N – 89 = 114571, N – 107 = 114553, N – 113 = 114547.
Now looking forward:
N + 11 = 114671 (isolate)
N + 19 = 114679 (isolate)
N + 29 = 114689, N + 31 = 114691 (twins)
Then we come to N + 53 = 114713 (isolate) in the next "century".
I will leave the later instances to be discovered. ;)

In brief:
A "pivotal" number in this neighbourhood is 114660 = 2*2*3*3*5*7*7*13.
The next prime after the twins 114689 & 114691 is
114,713

/RogerE :D

{Crystal ball prediction: N = 126126 will be of interest
when we get to it. No peeking! ;) }

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Re: Counting up in prime numbers #2

Post by BlackTuesday »

RogerE wrote:
21 Jul 2020 13:16
Wow! That last "century" was very prime-rich!
Before we move on, let's review that "century".
The recent posts found that there are 13 primes in the interval [114599, 114699],
of which 8 occur as twin primes, 2 as "near twins" and just 3 occur as "isolates".
That is a strikingly "dense" constellation of double stars.
.
.
Thanks for all the interesting info Roger! :)

Here is the next one:

114,743

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Re: Counting up in prime numbers #2

Post by RogerE »

That was 114660 + 83. That "heuristic" reasoning continues helpfully:

Next prime is 114660 + 89 =

114,749

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Re: Counting up in prime numbers #2

Post by BlackTuesday »

Next primes are near twins:

114,757

114,761

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Re: Counting up in prime numbers #2

Post by RogerE »

The near-twins just seen are translates of the near twins 97 and 101
114,757 = 114,660 + 97
114,761 = 114,660 + 101

Next we have another pair of near twins:
114,769
114,773


These near-twins are translates of the near twins 109 and 113
114,769 = 114,660 + 109
114,773 = 114,660 + 113

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Re: Counting up in prime numbers #2

Post by BlackTuesday »

Next prime is 114,660 + 121 =

114,781

An isolated prime for a change :D

Btw, nice new pro pic RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

Thanks BlackTuesday.

Notice that the latest prime is 114,781 = 114,660 + 121, where the offset is 11^2,
the square of the smallest prime not present in 114,660 = 2x2x3x3x5x7x7x13. :D

Next we have twin primes again!

114,797
114,799

Here the offset from 114,660 is the pair of twin primes 137 and 139.

/RogerE :D

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Re: Counting up in prime numbers #2

Post by BlackTuesday »

Next prime is 114,660 + 149 =

114,809

An isolated prime again :)

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Re: Counting up in prime numbers #2

Post by RogerE »

Yes, 114,809 = 114,660 + 149, with prime offset 149. Crystal ball says:
any further primes p < 114,660 + 187 will be of the form 114,660 + q,
where q is a prime, since 187 = 11x17 is the next possible composite offset
(after 121 = 11^2) sharing no prime factor with 114,660.

114,827

This has offset 167, which is prime. :)

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Re: Counting up in prime numbers #2

Post by BlackTuesday »

And the next one is

114,833

an offset of 173, which is a prime :D

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Re: Counting up in prime numbers #2

Post by RogerE »

Yes, BlackTuesday, you're seeing the underlying structure :D

114,847

This has the composite offset predicted:
114660 +187, where 187 = 11x17, the product of the two smallest primes not factors of 114660. :D

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Re: Counting up in prime numbers #2

Post by BlackTuesday »

Next one is

114,859

an offset of 199, which is a prime again :)

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Re: Counting up in prime numbers #2

Post by RogerE »

114,883

= 114,660 + 223, prime offset :D

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Re: Counting up in prime numbers #2

Post by BlackTuesday »

The next one is 114,660 + 229 =

114,889

a prime offset again :D

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Re: Counting up in prime numbers #2

Post by RogerE »

We have already passed the composite 11x19 = 209 as a possible offset.
The next possible composite offset is 11x23 = 253. Until then offsets must be prime.

Meanwhile, back on the thread, we enter a new century:
114,901

Offset 241 = p.

/RogerE :D

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Re: Counting up in prime numbers #2

Post by BlackTuesday »

Aaaaand, here comes the next one:

114,913

which is an offset of 253, not a prime this time ;)

but 253 = 11*23, both of which are p :D

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Re: Counting up in prime numbers #2

Post by RogerE »

That composite offset was mentioned in the previous post.
The next possible composite offsets from 114,660 are
17•17 = 289, 11x29 = 319 and 17x19 = 323.

114,913 is followed by:

114,941

Offset 281 = p.

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Re: Counting up in prime numbers #2

Post by BlackTuesday »

Next one comes as:

114,967

an offset of 307, a prime :)

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Re: Counting up in prime numbers #2

Post by RogerE »

114,973

Offset 313 = p.

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Re: Counting up in prime numbers #2

Post by BlackTuesday »

Twin primes after quite some time :)

114,997

115,001


Offsets this time are 337 and 341

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Re: Counting up in prime numbers #2

Post by RogerE »

The next possible composite offsets from 114,660 are
11x31 = 341, 19x19 = 361, 17x23 = 391, 11x37 = 407.
Previous BlackTuesday post shows 11x31 succeeds, along with p = 337.

115,013

Offset 353 = p.

/RogerE :D

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Re: Counting up in prime numbers #2

Post by BlackTuesday »

You're correct, and twin primes again :D

115019

115021


Offsets are 359 and 361 ... first one prime, second one as you told before :)

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Re: Counting up in prime numbers #2

Post by RogerE »

Thanks BlackTuesday :D

Let's review what we have observed: Recall that 114,660 = 2x2x3x3x5x7x7x13

Since 114,660 was recognised as being "pivotal" in the neighbourhood
where we're hunting for primes, it has served to explain why the "next"
primes are 114,660 + d with d equal to a prime (11 or at least 17) or else
a composite with prime factors from that set. In fact, the composites
will have exactly two prime factors (possibly equal) until d reaches at
least as far as 11^3 = 1331.

Hence the calculations that the next potential composite values of d
are 19x19 = 361, 17x23 = 391, 11x37 = 407, 19x23 = 437, etc

Lest we think these observations explain "everything", I would like to
point out that they only show us the set of potential values of d.
At every step we still have to determine whether a potential value does
or does not produce the next prime in our list.

For example, the prime offset 367 does not yield a new prime for our list,
because 114,660 + 367 is divisible by 11. The prime offset 373 does not yield
a new prime for our list, because 114,660 + 373 is divisible by 37. And so on...

The crystal ball predicts that 120,120 will be a nice pivotal number when
we reach its neighbourhood.
___________________________
.
Meanwhile, back at the ranch, the next primes are two near-twins:

115,057
115,061

Offsets 397 and 401, themselves near-twin primes.

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

After two days:

115,067

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Re: Counting up in prime numbers #2

Post by BlackTuesday »

Back! :) .... Thanks for the great analysis Roger! ... sometimes I think may be you're a computer :D

Next one is:

115,079

A prime offset of 419 :)

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