Thanks RogerE!

And guess what, twin primes again

**114,641**

114,643

114,643

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Thanks RogerE!

And guess what, twin primes again

114,643

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Excellent! This neighbourhood is quite crowded...

**114,649**

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After almost 3.5 days:

TWIN PRIMES!

**114,659**

**114,661**

/RogerE

TWIN PRIMES!

/RogerE

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Twin primes again

**114,689**

114,691

114,691

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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

{Crystal ball prediction: N = 126126 will be of interest

when we get to it. No peeking! }

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

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

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

/RogerE

{Crystal ball prediction: N = 126126 will be of interest

when we get to it. No peeking! }

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Thanks for all the interesting info Roger!RogerE wrote: ↑21 Jul 2020 13:16Wow! 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 the next one:

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

Next prime is 114660 + 89 =

**114,749**

Next prime is 114660 + 89 =

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Next primes are near twins:

**114,757**

114,761

114,761

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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

114,757 = 114,660 + 97

114,761 = 114,660 + 101

Next we have another pair of near twins:

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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Next prime is 114,660 + 121 =

**114,781**

An isolated prime for a change

Btw, nice new pro pic RogerE

An isolated prime for a change

Btw, nice new pro pic RogerE

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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.

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

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.

Next we have twin primes again!

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

/RogerE

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Next prime is 114,660 + 149 =

**114,809**

An isolated prime again

An isolated prime again

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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.

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.

This has offset 167, which is prime.

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And the next one is

**114,833**

an offset of 173, which is a prime

an offset of 173, which is a prime

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Yes, **BlackTuesday**, you're seeing the underlying structure

**114,847**

This has the composite offset predicted:

114660 +187, where 187 = 11x17, the product of the two smallest primes not factors of 114660.

This has the composite offset predicted:

114660 +187, where 187 = 11x17, the product of the two smallest primes not factors of 114660.

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Next one is

**114,859**

an offset of 199, which is a prime again

an offset of 199, which is a prime again

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= 114,660 + 223, prime offset

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The next one is 114,660 + 229 =

**114,889**

a prime offset again

a prime offset again

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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

The next possible composite offset is 11x23 = 253. Until then offsets must be prime.

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

Offset 241 = p.

/RogerE

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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

which is an offset of

but

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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.

The next possible composite offsets from 114,660 are

17•17 = 289, 11x29 = 319 and 17x19 = 323.

114,913 is followed by:

Offset 281 = p.

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Next one comes as:

**114,967**

an offset of 307, a prime

an offset of 307, a prime

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Offset 313 = p.

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Twin primes after quite some time

114,997

115,001

Offsets this time are 337 and 341

114,997

115,001

Offsets this time are 337 and 341

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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

11x31 = 341, 19x19 = 361, 17x23 = 391, 11x37 = 407.

Previous

Offset 353 = p.

/RogerE

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You're correct, and twin primes again

**115019**

115021

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

115021

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

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Thanks **BlackTuesday **

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.

**115,057**

**115,061**

Offsets 397 and 401, themselves near-twin primes.

/RogerE

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

Since

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

At every step we

or does not produce the next prime in our list.

For example, the prime offset 367 does

because 114,660 + 367 is divisible by 11. The prime offset 373 does

a new prime for our list, because 114,660 + 373 is divisible by 37. And so on...

The crystal ball predicts that

we reach its neighbourhood.

___________________________

.

Meanwhile, back at the ranch, the next primes are two near-twins:.

Offsets 397 and 401, themselves near-twin primes.

/RogerE

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After two days:

**115,067**

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

Next one is:

**115,079**

A prime offset of 419

Next one is:

A prime offset of 419

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Hello **BlackTuesday,** I like your new avatar! (Does that prove I'm not a computer‽)

**115,099**

Prime offset p = 599.

/RogerE

Prime offset p = 599.

/RogerE

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Hehehe, thank you! ... also, I doubt if a computer will ever be able to be a stamp collector

I'm surprised though, what is your prime offset doing so far ahead of the lane

115,117

A prime offset of 457

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Thanks for the "offset" correction **BlackTuesday.**

After 115,117 we have

**115,123**

**115,127**

Near twin primes, offsets coming from the later two members

of the maximal prime configuration 461, 463, 467.

Notice that offset 461 —> 115,121 = 19x73x83.

/RogerE

[Later correction: p = 439, of course! Another proof I'm not a computer ]

After 115,117 we have

Near twin primes, offsets coming from the later two members

of the maximal prime configuration 461, 463, 467.

Notice that offset 461 —> 115,121 = 19x73x83.

/RogerE

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You're welcome Roger!

Next one is:

**115,133**

An offset of 473 = 11 * 43

Next one is:

An offset of 473 = 11 * 43

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Twin primes!

**115,151**

**115,153**

Prime offset p = 491, then composite offset q = 493 = 17*29

/RogerE

Prime offset p = 491, then composite offset q = 493 = 17*29

/RogerE

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Solo prime:

**115,163**

And a prime offset p = 503

And a prime offset p = 503

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Solo prime:

**115,183**

And a prime offset p = 523

And a prime offset p = 523

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Solo prime again:

115,201

And a prime offset p = 541

115,201

And a prime offset p = 541

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Solo prime:

**115,211**

And a COMPOSITE offset q = 19x29

And a COMPOSITE offset q = 19x29

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Solo prime:

**115,223**

And a prime offset p = 563

And a prime offset p = 563

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Solo prime:

**115,237**

And a prime offset p = 577

And a prime offset p = 577

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Solo prime again:

**115,249**

And a composite offset q = 589 = 19*31

And a composite offset q = 589 = 19*31

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Solo prime again:

**115,259**

And a prime offset p = 599

And a prime offset p = 599

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Waited a couple of days...

Solo prime again:

**115,279**

And a prime offset p = 619

Solo prime again:

And a prime offset p = 619

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25hrs later:

TWIN primes:

**115,301**

**115,303**

And twin primes offset p = 641 & 643

TWIN primes:

And twin primes offset p = 641 & 643

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Waited several days...

Solo prime again:

**115,309**

/RogerE

Solo prime again:

/RogerE

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Where are the others who contribute to this thread?

Waited several days...

Twin primes:

**115,319**

**115,321**

/RogerE

Waited several days...

Twin primes:

/RogerE

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Waited two days...

Near twin primes

(maximum number of primes within 5 consecutive numbers):

**115,327**

**115,331**

/RogerE

Near twin primes

(maximum number of primes within 5 consecutive numbers):

/RogerE

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After two days...

Isolated prime:

**115,337**

/RogerE

Isolated prime:

/RogerE

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