Counting up in prime numbers #2

BlackTuesday · Post by **BlackTuesday** » 03 Nov 2020 07:47

Thanks Roger!

And an isolated prime again:

116,027

116,027 = 114660 + 1367, and offset 1367 = prime

RogerE · Post by **RogerE** » 03 Nov 2020 10:29

Another isolated prime:

116,041

116,041 = 114,660 + 1381, with offset 1381 = prime

/RogerE

BlackTuesday · Post by **BlackTuesday** » 04 Nov 2020 08:50

And another isolated prime:

116,047

116,047 = 114,660 + 1387, with offset 1387 = 19x73 both of which are prime

RogerE · Post by **RogerE** » 04 Nov 2020 10:14

Another isolated prime:

116,089

116,089 = 114,660 + 1429, with offset 1429 = prime

/RogerE

RogerE · Post by **RogerE** » 10 Nov 2020 00:53

Next we have twin primes:

116,099
116,101

116,099 = 114,660 + 1439, with offset 1439 = prime
116,101 = 114,660 + 1441, with offset 1441 = composite, 11 x 131

/RogerE

RogerE · Post by **RogerE** » 12 Nov 2020 02:09

An isolated prime (but not too isolated — prime real estate nearby
on either side...

116,107

116,107 = 114,660 + 1447, with offset 1447 = prime

/RogerE

RogerE · Post by **RogerE** » 14 Nov 2020 22:47

An isolated prime:

116,113

116,113 = 114,660 + 1453, with offset 1453 = prime

/RogerE

BlackTuesday · Post by **BlackTuesday** » 15 Nov 2020 10:42

Back!

An isolated prime again:

116,131

116,131 = 114,660 + 1471, with offset 1471 = prime

RogerE · Post by **RogerE** » 15 Nov 2020 10:46

Welcome back, BlackTuesday !

An isolated prime again:

116,141

116,141 = 114,660 + 1481, with offset 1481 = prime

BlackTuesday · Post by **BlackTuesday** » 16 Nov 2020 14:45

Thanks Roger!

And an isolated prime again:

116,159

116,159 = 114,660 + 1499, with offset 1499 = prime

RogerE · Post by **RogerE** » 17 Nov 2020 01:52

Another isolated prime:

116,167

116,167 = 114,660 + 1507, with composite offset 1507 = 11 x 137.

/RogerE

BlackTuesday · Post by **BlackTuesday** » 17 Nov 2020 14:00

Another isolated prime:

116,177

116,177 = 114,660 + 1517, and offset 1517 = composite of 37 x 41

RogerE · Post by **RogerE** » 17 Nov 2020 17:33

A note about the previous offset, 1517 = composite of 37 x 41

1517 = 1521 - 4 = 39^2 - 2^2, a difference of two squares.
Since a^2 - b^2 = (a - b) x (a + b), we can always factorise such a number

In this instance, 1517 = (39 - 2) x (39 + 2) = 37 x 41.

A related note: 40^2 - 39^2 = 40 + 39, so 39^2 = 1600 - 79 = 1521

_____________________________________________
Now for the next primes, a pair of twins.

116,189
116,191

116,189 = 114,660 + 1529, and offset 1529 = composite of 11 x 139
116,191 = 114,660 + 1531, and offset 1531 = prime

/RogerE

BlackTuesday · Post by **BlackTuesday** » 19 Nov 2020 08:03

Thanks for the interesting notes, Roger!

Next is another isolated prime:

116,201

116,201 = 114,660 + 1541, and offset 1541 = composite of 23 x 67

RogerE · Post by **RogerE** » 20 Nov 2020 01:42

Next a big gap, then a pair of near twins:

116,239
116,243

116,239 = 114,660 + 1579, and offset 1579 is prime
116,243 = 114,660 + 1583, and offset 1583 is prime

/RogerE

BlackTuesday · Post by **BlackTuesday** » 21 Nov 2020 02:11

Next is an isolated prime:

116,257

116,257 = 114,660 + 1597, and offset 1597 = prime

RogerE · Post by **RogerE** » 21 Nov 2020 03:37

Next another pair of near twin primes:

116,269
116,273

The offsets this time are also a pair of near twin primes:
116,269 = 114,660 + 1609, and offset 1609 = prime
116,273 = 114,660 + 1613, and offset 1613 = prime

/RogerE

BlackTuesday · Post by **BlackTuesday** » 22 Nov 2020 07:53

Next is an isolated prime:

116,279

116,279 = 114,660 + 1619, and offset 1619 = prime

RogerE · Post by **RogerE** » 22 Nov 2020 11:37

Next is a very isolated prime:

116,293

116,293 = 114,660 + 1633, and composite offset 1633 = 23 x 71

/RogerE

RogerE · Post by **RogerE** » 25 Nov 2020 01:54

Next is an isolated prime:

116,329

116,329 = 114,660 + 1669, and offset 1669 = prime

/RogerE

RogerE · Post by **RogerE** » 26 Nov 2020 09:53

Next is an isolated prime:

116,341

116,341 = 114,660 + 1681, and offset 1681 = perfect square 41^2

A reminder of why 114,660 is "pivotal" in this region:
114,660 = 2x2x3x3x5x7x7x13
It contains a nice cluster of early primes, so the offsets to prime neighbours of 114,660 must be
composed of "larger" primes from among 11, 17, 19, ... so offset numbers are "relatively scarce".

/RogerE

BlackTuesday · Post by **BlackTuesday** » 27 Nov 2020 07:39

Thanks for the interesting info Roger!

Next is again an isolated prime:

116,351

116,351 = 114,660 + 1691, and offset 1691 = composite of 19 x 89

RogerE · Post by **RogerE** » 27 Nov 2020 09:12

Another isolated prime:

116,359

116,359 = 114,660 + 1699, and offset 1699 = prime

RogerE · Post by **RogerE** » 29 Nov 2020 03:18

Another isolated prime:

116,371

116,371 = 114,660 + 1711, and composite offset 1711 = 29x59

[1711 = 1936 – 225 = 44^2 - 15^2 = (44 - 15)x(44 + 15) = 29x59
If I remember correctly, factorising ad a difference of squares
was a method used by Fermat.]

RogerE · Post by **RogerE** » 29 Nov 2020 23:42

Another isolated prime:

116,381

116,381 = 114,660 + 1721, and offset 1721 = prime

[Correcting some "fat fingered" typing In my previous post, I meant
to say: if I remember correctly, factorising as a difference of squares
was a method used by Fermat.]

RogerE · Post by **RogerE** » 03 Dec 2020 04:09

Another isolated prime:

116,387

116,387 = 114,660 + 1727, and offset 1727 = 11x157

/RogerE

P.S. BlackTuesday, I think you might like
https://www.stampboards.com/viewtopic.php?f=11&t=20547&p=6939502&hilit=fibonacci#p6939502

RogerE · Post by **RogerE** » 07 Dec 2020 13:44

Another isolated prime:

116,411

116,411 = 114,660 + 1751, and offset 1751 = 17x103

/RogerE

P.S. BlackTuesday, I think you might like
https://www.stampboards.com/viewtopic.php?f=11&t=20547&p=6939502&hilit=fibonacci#p6939502

RogerE · Post by **RogerE** » 08 Dec 2020 00:42

A theoretical note

A key post in this thread occurred at
https://www.stampboards.com/viewtopic.php?f=11&t=85523&start=502

RogerE wrote: ↑
19 Jul 2020 12:21

TWIN PRIMES!

114,659
114,661

Primes in the neighbourhood of N

This provided an incentive to look at the intervening number N = 114,660 and the offsets
from this number to the neighbouring primes, listed in this thread. Nearby primes were seen
to cluster here, and be "almost" symmetrically located around N.

It turns out that N is composed of small primes: N = 2x2x3x3x5x7x7x13, so if q is a composite number in the range 1< q < 11x11 = 121, it will share a prime factor with N, and then N – q and N + q will both be multiples of that shared prime factor. Therefore, if N – q or N + q is to be prime for offset 1 < q < 121, then q is necessarily prime. (Then, even if N – q or N + q is not prime, its prime factors can only include primes 11 or p ≥ 17).

Here is a table showing what actually occurs for small offsets:

.

This accounts for all the primes in the immediate neighbourhood of N, and
the principle continues to "explain" which larger offsets q match prime N + q.

A glance to the future

Delete the factor 3x7 = 21 from N, and replace it by 2x11 = 22, to form the
modestly larger number M = 2x2x2x3x5x7x11x13 = 120,120. The prime factors
of M are all primes below 17, so M – q or M + q will necessarily be composite
unless the offset q is 1 or has no prime factor less than 17. In particular, if
1 < q < 17^2 = 289 then M – q or M + q can only be prime of q is prime.

In due course we shall be examining numbers in the neighbourhood of 120,120
in this thread...

/RogerE

RogerE · Post by **RogerE** » 08 Dec 2020 01:00

Another isolated prime:

116,423

116,423 = 114,660 + 1763, and composite offset 1763 = 41x43
[Note:1763 = 1764 – 1 = 42^2 – 1^2 = (42 – 1)(42 + 1) = 41x43.]

A glance to the future:

116,423 = 120,120 – 3697, and offset 3697 = prime.

/RogerE

RogerE · Post by **RogerE** » 09 Dec 2020 02:38

Another isolated prime:

116,437

116,437 = 114,660 + 1777, and offset 1777 = prime

A glance to the future:

116,437 = 120,120 – 3683, and offset 3683 = 29x127.
(I was hoping for a product of three primes, but not this time!)

/RogerE

RogerE · Post by **RogerE** » 10 Dec 2020 00:49

A pair of near-twin primes:

116,443
116,447

116,443 = 114,660 + 1783, and offset 1783 = prime
116,447 = 114,660 + 1787, and offset 1787 = prime

A glance to the future:

116,443 = 120,120 – 3677, and offset 3683 = prime.
116,447 = 120,120 – 3673, and offset 3683 = prime.

I should have thought more carefully about my previous footnote.
The smallest possible product of three primes which can be an
offset of a prime from 120,120 is 17^3 = 4913, and we're already
much closer to 120,120 than that!

/RogerE

RogerE · Post by **RogerE** » 10 Dec 2020 08:26

An isolated prime:

116,461

116,461 = 114,660 + 1801, and offset 1801 = prime

A glance to the future (with corrected mistyping from prior post):

116,443 = 120,120 – 3677, and offset 3677 = prime.
116,447 = 120,120 – 3673, and offset 3673 = prime.
116,461 = 120,120 – 3673, and offset 3659 = prime.

/RogerE

RogerE · Post by **RogerE** » 14 Dec 2020 02:22

After 90hrs+

An isolated prime:

116,471

116,471 = 114,660 + 1811, and offset 1811 = prime

A glance to the future:
116,471 = 120,120 – 3649, and offset 3659 = prime.

/RogerE

RogerE · Post by **RogerE** » 16 Dec 2020 10:35

An isolated prime:

116,483

116,483 = 114,660 + 1823, and offset 1823 = prime

Also:
116,483 = 120,120 – 3637, and offset 3637 = prime.
I will omit further discussion of offsets from 120,120
for the, next primes until we pass 118,000...

/RogerE

RogerE · Post by **RogerE** » 20 Dec 2020 01:16

An isolated prime:

116,491

116,491 = 114,660 + 1831, and offset 1831 = prime

/RogerE

RogerE · Post by **RogerE** » 26 Dec 2020 03:24

An isolated prime:

116,507

116,507 = 114,660 + 1847, and offset 1847 = prime

/RogerE

RogerE · Post by **RogerE** » 26 Dec 2020 18:45

A prime cluster — two pairs of twin primes as close as possible:
four primes in five consecutive odd numbers

116,531
116,533
116,537
116,539

116,535 = 3x5x17x457

116,531 = 114,660 + 1871, and offset 1871 = prime
116,533 = 114,660 + 1873, and offset 1873 = prime
116,537 = 114,660 + 1877, and offset 1877 = prime
116,539 = 114,660 + 1879, and offset 1879 = prime

The offsets 1871, 1873, 1877, 1879 are a corresponding configuration
of four primes in five consecutive odd numbers

1875 = 3x5x5x5x5

/RogerE

RogerE · Post by **RogerE** » 31 Dec 2020 12:38

Looks like BlackTuesday is still on mini-sabbatical..,
but Happy New Year in any case

An isolated prime

116,549

116,549 = 114,660 + 1889, and offset 1889 = prime

/RogerE

RogerE · Post by **RogerE** » 03 Jan 2021 00:08

An isolated prime

116,579

116,579 = 114,660 + 1919, and offset 1919 = 19x101

/RogerE

RogerE · Post by **RogerE** » 04 Jan 2021 18:12

An isolated prime

116,593

116,593 = 114,660 + 1933, and offset 1933 = prime

/RogerE

RogerE · Post by **RogerE** » 06 Jan 2021 02:25

An isolated prime

116,639

116,639 = 114,660 + 1979, and offset 1979 = prime

/RogerE

RogerE · Post by **RogerE** » 06 Jan 2021 12:06

An isolated prime

116,657

116,657 = 114,660 + 1997, and offset 1997 = prime

/RogerE

RogerE · Post by **RogerE** » 09 Jan 2021 03:05

An isolated prime:

116,663

116,663 = 114,660 + 2003, and offset 2003 = prime

/RogerE

RogerE · Post by **RogerE** » 12 Jan 2021 14:54

Still hoping for reinforcements...

An isolated prime:

116,681

116,681 = 114,660 + 2021, and offset 2021 = 43x47

Difference of squares method:
2021 = 2025 - 4 = 45^2 - 2^2 = (45 - 2)x(45 + 2) = 43x47

/RogerE