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### Re: Counting up in prime numbers #2

Posted: **03 Nov 2020 07:47**

by **BlackTuesday**

Thanks

Roger!

And an isolated prime again:

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

### Re: Counting up in prime numbers #2

Posted: **03 Nov 2020 10:29**

by **RogerE**

Another isolated prime:

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

/RogerE

### Re: Counting up in prime numbers #2

Posted: **04 Nov 2020 08:50**

by **BlackTuesday**

And another isolated prime:

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

### Re: Counting up in prime numbers #2

Posted: **04 Nov 2020 10:14**

by **RogerE**

Another isolated prime:

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

/RogerE

### Re: Counting up in prime numbers #2

Posted: **10 Nov 2020 00:53**

by **RogerE**

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

### Re: Counting up in prime numbers #2

Posted: **12 Nov 2020 02:09**

by **RogerE**

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

### Re: Counting up in prime numbers #2

Posted: **14 Nov 2020 22:47**

by **RogerE**

An isolated prime:

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

/RogerE

### Re: Counting up in prime numbers #2

Posted: **15 Nov 2020 10:42**

by **BlackTuesday**

Back!

An isolated prime again:

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

### Re: Counting up in prime numbers #2

Posted: **15 Nov 2020 10:46**

by **RogerE**

Welcome back,

**BlackTuesday** !

An isolated prime again:

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

### Re: Counting up in prime numbers #2

Posted: **16 Nov 2020 14:45**

by **BlackTuesday**

Thanks

Roger!

And an isolated prime again:

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

### Re: Counting up in prime numbers #2

Posted: **17 Nov 2020 01:52**

by **RogerE**

Another isolated prime:

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

/RogerE

### Re: Counting up in prime numbers #2

Posted: **17 Nov 2020 14:00**

by **BlackTuesday**

Another isolated prime:

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

### Re: Counting up in prime numbers #2

Posted: **17 Nov 2020 17:33**

by **RogerE**

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

### Re: Counting up in prime numbers #2

Posted: **19 Nov 2020 08:03**

by **BlackTuesday**

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

### Re: Counting up in prime numbers #2

Posted: **20 Nov 2020 01:42**

by **RogerE**

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

### Re: Counting up in prime numbers #2

Posted: **21 Nov 2020 02:11**

by **BlackTuesday**

Next is an isolated prime:

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

### Re: Counting up in prime numbers #2

Posted: **21 Nov 2020 03:37**

by **RogerE**

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

### Re: Counting up in prime numbers #2

Posted: **22 Nov 2020 07:53**

by **BlackTuesday**

Next is an isolated prime:

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

### Re: Counting up in prime numbers #2

Posted: **22 Nov 2020 11:37**

by **RogerE**

Next is a

*very* isolated prime:

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

/RogerE

### Re: Counting up in prime numbers #2

Posted: **25 Nov 2020 01:54**

by **RogerE**

Next is an isolated prime:

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

/RogerE

### Re: Counting up in prime numbers #2

Posted: **26 Nov 2020 09:53**

by **RogerE**

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

### Re: Counting up in prime numbers #2

Posted: **27 Nov 2020 07:39**

by **BlackTuesday**

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

### Re: Counting up in prime numbers #2

Posted: **27 Nov 2020 09:12**

by **RogerE**

Another isolated prime:

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

### Re: Counting up in prime numbers #2

Posted: **29 Nov 2020 03:18**

by **RogerE**

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.]

### Re: Counting up in prime numbers #2

Posted: **29 Nov 2020 23:42**

by **RogerE**

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.]

### Re: Counting up in prime numbers #2

Posted: **03 Dec 2020 04:09**

by **RogerE**

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

### Re: Counting up in prime numbers #2

Posted: **07 Dec 2020 13:44**

by **RogerE**

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

### Re: Counting up in prime numbers #2

Posted: **08 Dec 2020 00:42**

by **RogerE**

**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

### Re: Counting up in prime numbers #2

Posted: **08 Dec 2020 01:00**

by **RogerE**

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

### Re: Counting up in prime numbers #2

Posted: **09 Dec 2020 02:38**

by **RogerE**

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

### Re: Counting up in prime numbers #2

Posted: **10 Dec 2020 00:49**

by **RogerE**

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

### Re: Counting up in prime numbers #2

Posted: **10 Dec 2020 08:26**

by **RogerE**

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

### Re: Counting up in prime numbers #2

Posted: **14 Dec 2020 02:22**

by **RogerE**

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

### Re: Counting up in prime numbers #2

Posted: **16 Dec 2020 10:35**

by **RogerE**

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

### Re: Counting up in prime numbers #2

Posted: **20 Dec 2020 01:16**

by **RogerE**

An isolated prime:

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

/RogerE

### Re: Counting up in prime numbers #2

Posted: **26 Dec 2020 03:24**

by **RogerE**

An isolated prime:

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

/RogerE

### Re: Counting up in prime numbers #2

Posted: **26 Dec 2020 18:45**

by **RogerE**

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

### Re: Counting up in prime numbers #2

Posted: **31 Dec 2020 12:38**

by **RogerE**

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

### Re: Counting up in prime numbers #2

Posted: **03 Jan 2021 00:08**

by **RogerE**

An isolated prime

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

/RogerE

### Re: Counting up in prime numbers #2

Posted: **04 Jan 2021 18:12**

by **RogerE**

An isolated prime

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

/RogerE

### Re: Counting up in prime numbers #2

Posted: **06 Jan 2021 02:25**

by **RogerE**

An isolated prime

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

/RogerE

### Re: Counting up in prime numbers #2

Posted: **06 Jan 2021 12:06**

by **RogerE**

An isolated prime

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

/RogerE

### Re: Counting up in prime numbers #2

Posted: **09 Jan 2021 03:05**

by **RogerE**

An isolated prime:

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

/RogerE

### Re: Counting up in prime numbers #2

Posted: **12 Jan 2021 14:54**

by **RogerE**

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