Counting up in prime numbers #2

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Re: Counting up in prime numbers #2

Post by RogerE »

Pair of twin primes:

116,789
116,791

Offsets, also a pair of twin primes:
116,789 = 114,660 + 2129, and offset 2129 = prime
116,791 = 114,660 + 2131, and offset 2131 = prime

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

Isolated prime:

116,797

Offset:
116,797 = 114,660 + 2137, and offset 2137 = prime

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

Isolated prime:

116,803

Offset:
116,803 = 114,660 + 2143, and offset 2143 = prime

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

Isolated prime:

116,819

Offset:
116,819 = 114,660 + 2159, and composite offset 2159 = 17 x 127

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

BlackTuesday and Wolfgang, missing you on this thread... :(


Isolated prime:

116,827

Offset:
116,827 = 114,660 + 2167, and composite offset 2167 = 11 x 197

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

Isolated prime:

116,833

Offset:
116,833 = 114,660 + 2173, and composite offset 2173 = 41 x 53

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

Isolated prime:

116,849

Offset:
116,849 = 114,660 + 2189, and composite offset 2189 = 11 x 199

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

Isolated prime:

116,867

Offset:
116,867 = 114,660 + 2207, and offset 2207 = prime

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

Isolated prime:

116,881

Offset:
116,881 = 114,660 + 2221, and offset 2221 = prime

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

Isolated prime:

116,903

Offset:
116,903 = 114,660 + 2243, and offset 2243 = prime

/RogerE :D
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Re: Counting up in prime numbers #2

Post by RogerE »

I hope some former contributors to this thread will reappear soon...
or maybe a brave new prime enthusiast will surprise us with a post ;)

Isolated prime:

116,911

Offset:
116,911 = 114,660 + 2251, and offset 2251 = prime

/RogerE :D
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Re: Counting up in prime numbers #2

Post by RogerE »

A dense patch of primes — 4 primes in 6 consecutive odd numbers
(maximum density possible beyond 3, since two multiples of 3 must
occur among six consecutive odd numbers).

116,923
116,927
116,929
116,933

Offsets:
116,923 = 114,660 + 2263, and offset 2263 = 31 x 73
116,927 = 114,660 + 2267, and offset 2267 = prime
116,929 = 114,660 + 2269, and offset 2269 = prime
116,933 = 114,660 + 2273, and offset 2273 = prime

/RogerE :D
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Re: Counting up in prime numbers #2

Post by RogerE »

Isolated prime:

116,953

Offset:
116,953 = 114,660 + 2293, and offset 2293 = prime

/RogerE :D
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Re: Counting up in prime numbers #2

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Isolated prime:

116,959

Offset:
116,959 = 114,660 + 2299, and composite offset 2299 = 11 x 209

/RogerE :D
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Re: Counting up in prime numbers #2

Post by RogerE »

Isolated prime:

116,969

Offset:
116,969 = 114,660 + 2309, and offset 2309 = prime[343]

I found a website which reports the rank of a given small prime number.
It reports that 2309 is the 343rd prime, but it only confirms that 116969
is a prime, without telling its rank...
Incidentally, 343 = 7^3.

In Dec 2020 I added a post in this thread discussing the clustering of primes
around N = 114,660, and noting that M = 120,120 would be another cluster
point we could watch out for in future. The discussion was at
https://www.stampboards.com/viewtopic.php?f=11&t=85523&p=6949208&#p6949208

It was noted that if a prime has composite offset from 120120, the prime factors
of the composite offset must be at least as large as 17. (Indeed, if the absolute
value of the offset is q, and 1 < q < 289 = 17^2, then q must be prime.)

We could now look at the offset of 116969 from 120120:
116,969 = 120,120 - 3151 and the offset is composite: 3151 = 23 x 137.

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

Isolated prime:

116,981

Offset:
116,981 = 114,660 + 2321, and offset is composite: 2321 = 11x211

We could also look at the offset of 116981 from 120120:
116,981 = 120,120 – 3139 and offset is composite: 3139 = 43 x 73.

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

A pair of near primes, but not twins:

116,989
116,993

Offset:
116,989 = 114,660 + 2329, and offset is composite: 2329 = 17 x 137.
116,9893 = 114,660 + 2333, and offset is prime: 2333 = P(345).

We could also look at the offsets from 120120:
116,989 = 120,120 – 3131 and offset is composite: 3131 = 31 x 101.
116,993 = 120,120 – 3127 and offset is composite: 3127 = 53 x 59.

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

A lone prime:

117,017

Offset:
117,017 = 114,660 + 2357, and offset is prime: 2357 = P(348)
"2357 is the smallest prime that contains all of the prime digits."


The offset from 120120:
117,017 = 120,120 – 3103 and offset is composite: 3103 = 29 x 107.

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

A lone prime:

117,023

Offset:
117,023 = 114,660 + 2363, and offset is composite: 2363 = 17 x 139

The offset from 120120:
117,023 = 120,120 – 3097 and offset is composite: 3097 = 19 x 163.

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

A maximal prime cluster (three primes in four consecutive odd numbers):

117,037
117,041
117,043

Offset:
117,037 = 114,660 + 2377, and offset is prime: 2377 = P(352)
117,041 = 114,660 + 2381, and offset is prime: 2381 = P(353)
117,043 = 114,660 + 2383, and offset is prime: 2383 = P(354)

The offset from 120120:
117,037 = 120,120 – 3083 and offset is prime: 3083 = P(441).
117,041 = 120,120 – 3079 and offset is prime: 3079 = P(440).
117,043 = 120,120 – 3077 and offset is composite: 3077 = 17 x 181.

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

A lone prime:

117,053

Offset:
117,053 = 114,660 + 2393, and offset is prime: 2393 = P(356)

The offset from 120120:
117,053 = 120,120 – 3067, and offset is prime: 3067 = P(439)

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

A lone prime:

117,071

Offset:
117,071 = 114,660 + 2411, and offset is prime: 2411 = P(358)

The offset from 120120:
117,071 = 120,120 – 3049, and offset is prime: 3049 = P(437)

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

A lone prime:

117,101

Offset:
117,101 = 114,660 + 2441, and offset is prime: 2441 = P(362)

The offset from 120120:
117,101 = 120,120 – 3019, and offset is prime: 3019 = P(433)

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

A lone prime:

117,109

Offset:
117,109 = 114,660 + 2449, and offset is composite: 2449 = 31 x 79

The offset from 120120:
117,109 = 120,120 – 3011, and offset is prime: 3011 = P(432)

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

A lone prime:

117,119

Offset:
117,119 = 114,660 + 2459, and offset is prime: 2459 = P(364)

The offset from 120120:
117,119 = 120,120 – 3001, and offset is prime: 3001 = P(431)

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

This page is full of Major Mitchell avatars — where are past contributors now?

A lone prime:

117,127

Offset:
117,127 = 114,660 + 2467, and offset is prime: 2467 = P(365)

The offset from 120120:
117,127 = 120,120 – 2993, and offset is composite: 2993 = 41 x 73

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

A lone prime:

117,133

Offset:
117,133 = 114,660 + 2473, and offset is prime: 2473 = P(366)

The offset from 120120:
117,133 = 120,120 – 2987, and offset is composite: 2987 = 29 x 103

/RogerE :D

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Re: Counting up in prime numbers #2

Post by RogerE »

A pair of almost twin primes — among three odd numbers greater than 3, two primes is maximal:

117,163 and 117,167

Offset:
117,163 = 114,660 + 2503, and offset is prime: 2503 = P(368)
117,167 = 114,660 + 2507, and offset is composite: 2507 = 23 x 109

The offset from 120120:
117,163 = 120,120 – 2957, and offset is prime: 2957 = P(426)
117,167 = 120,120 – 2953, and offset is prime: 2953 = P(425)

/RogerE :D

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