And an isolated prime again:

**116,027**

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

**Moderator:** Volunteer Moderator Team

- BlackTuesday
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Thanks Roger!

And an isolated prime again:

**116,027**

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

And an isolated prime again:

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

Not totally absent, but due to workload currently in mini sabbatical mode on the board!

- RogerE
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Another isolated prime:

**116,041**

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

/RogerE

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

/RogerE

- BlackTuesday
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And another isolated prime:

**116,047**

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

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

Not totally absent, but due to workload currently in mini sabbatical mode on the board!

- RogerE
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Another isolated prime:

**116,089**

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

/RogerE

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

/RogerE

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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

116,099 = 114,660 + 1439, with offset 1439 = prime

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

/RogerE

- RogerE
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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

on either side...

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

/RogerE

- RogerE
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An isolated prime:

**116,113**

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

/RogerE

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

/RogerE

- BlackTuesday
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Back!

An isolated prime again:

116,131

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

An isolated prime again:

116,131

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

Not totally absent, but due to workload currently in mini sabbatical mode on the board!

- RogerE
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Welcome back, **BlackTuesday** !

An isolated prime again:

116,141

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

An isolated prime again:

116,141

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

- BlackTuesday
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Thanks Roger!

And an isolated prime again:

**116,159**

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

And an isolated prime again:

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

Not totally absent, but due to workload currently in mini sabbatical mode on the board!

- RogerE
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Another isolated prime:

**116,167**

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

/RogerE

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

/RogerE

- BlackTuesday
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Another isolated prime:

**116,177**

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

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

Not totally absent, but due to workload currently in mini sabbatical mode on the board!

- RogerE
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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

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 = 114,660 + 1529, and offset 1529 = composite of 11 x 139

116,191 = 114,660 + 1531, and offset 1531 = prime

/RogerE

- BlackTuesday
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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

Next is another isolated prime:

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

Not totally absent, but due to workload currently in mini sabbatical mode on the board!

- RogerE
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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

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

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

/RogerE

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Next is an isolated prime:

**116,257**

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

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

Not totally absent, but due to workload currently in mini sabbatical mode on the board!

- RogerE
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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

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
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Next is an isolated prime:

**116,279**

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

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

Not totally absent, but due to workload currently in mini sabbatical mode on the board!

- RogerE
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Next is a *very* isolated prime:

**116,293**

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

/RogerE

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

/RogerE

- RogerE
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Next is an isolated prime:

**116,329**

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

/RogerE

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

/RogerE

- RogerE
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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

116,341 = 114,660 + 1681, and offset 1681 =

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
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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

Next is again an isolated prime:

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

Not totally absent, but due to workload currently in mini sabbatical mode on the board!

- RogerE
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Another isolated prime:

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

- RogerE
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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.]

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
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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.]

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
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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

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

/RogerE

P.S.

https://www.stampboards.com/viewtopic.php?f=11&t=20547&p=6939502&hilit=fibonacci#p6939502

- RogerE
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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

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

/RogerE

P.S.

https://www.stampboards.com/viewtopic.php?f=11&t=20547&p=6939502&hilit=fibonacci#p6939502

- RogerE
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A key post in this thread occurred at

https://www.stampboards.com/viewtopic.php?f=11&t=85523&start=502

This provided an incentive to look at the intervening number

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:

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

the principle continues to "explain" which larger offsets q match prime N + q.

Delete the factor 3x7 = 21 from N, and replace it by 2x11 = 22, to form the

modestly larger number

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

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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

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
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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

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

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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

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

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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

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
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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

An isolated prime:

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
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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

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

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An isolated prime:

**116,491**

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

/RogerE

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

/RogerE

- RogerE
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An isolated prime:

**116,507**

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

/RogerE

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

/RogerE

- RogerE
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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

four primes in five consecutive odd numbers

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
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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

but

An isolated prime

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

/RogerE

- RogerE
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An isolated prime

**116,579**

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

/RogerE

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

/RogerE

- RogerE
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An isolated prime

**116,593**

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

/RogerE

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

/RogerE

- RogerE
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An isolated prime

**116,639**

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

/RogerE

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

/RogerE

- RogerE
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An isolated prime

**116,657**

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

/RogerE

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

/RogerE

- RogerE
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An isolated prime:

**116,663**

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

/RogerE

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

/RogerE

- RogerE
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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

An isolated prime:

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

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