Ten prime numbers describing an arithmetic sequence

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Discussion Overview

The discussion revolves around the identification of ten distinct prime numbers that form an arithmetic sequence, with a focus on their properties and the existence of such sequences among primes less than 3000 and potentially up to 1 million.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asserts that there exists a unique run of ten distinct primes below 3000 that form an arithmetic sequence.
  • Another participant questions the validity of the arithmetic properties of the primes, suggesting a contradiction based on the sums of the primes involved.
  • There are inquiries about the existence of runs of nine primes within the context of the ten primes and whether more runs exist beyond those identified.
  • Participants express interest in the longest run of arithmetically sequenced primes below 1 million and the conditions under which such runs can be found.
  • One participant mentions that the sequence can be expressed in the form p, p+d, p+2d..., where p is an odd prime and d is a positive integer, but raises concerns about the arithmetic implications of this formulation.

Areas of Agreement / Disagreement

Participants do not appear to reach consensus on the existence of additional runs of primes or the arithmetic properties discussed, indicating multiple competing views and unresolved questions.

Contextual Notes

There are limitations regarding the assumptions made about the properties of prime numbers in arithmetic sequences, and the discussion does not clarify the maximum bounds for finding additional runs of primes.

Who May Find This Useful

Mathematicians, number theorists, and enthusiasts interested in prime number sequences and their properties may find this discussion relevant.

K Sengupta
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Ten distinct prime numbers, each less than 3000, when arranged in increasing order of magnitude describe an arithmetic sequence.

What are these ten prime numbers?
 
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wrong
 
Last edited:
Jarle said:
But the sum of the prime numbers p and p+d are even, but also equal to the prime number p+3d, a contradiction.

Why p+p+d = p+3d ?
 
199 + n * 210 (where n = 0, 1, 2, ..., 9) are all primes.
I found this by brute force. Here's an easy question. This is a run of length 10. There are two runs of length 9. What are they?
 
Last edited:
jimmysnyder said:
Here's an easy question. This is a run of length 10. There are two runs of length 9. What are they?

Are you implying that there are more runs of 9 besides the 2 included within your run of 10?

But while we're doing brute force:
There's only 1 run of length 10 within all the primes up to a max of 3000. But how high do you have to push the max before you find another run of 10? How many runs of 10 are there with all the primes below 1 million?

DaveE
 
Jarle said:
They are on the form p,p+d,p+2d..., where p is an odd prime and d is a positive integer. But the sum of the prime numbers p and p+d are even, but also equal to the prime number p+3d, a contradiction.

p + p + d =/= p + d+d+d simple mistake
 
davee123 said:
Are you implying that there are more runs of 9 besides the 2 included within your run of 10?
This isn't jeopardy. Pose your answer in the form of an answer.
 
jimmysnyder said:
This isn't jeopardy. Pose your answer in the form of an answer.

Ok, just checking.

And another brute forcer: what's the longest run of arithmetically sequenced primes below 1 million?

DaveE
 
So, just so I don't leave these questions without answers:

There's only 1 run of length 10 within all the primes up to a max of 3000. But how high do you have to push the max before you find another run of 10?

53813

How many runs of 10 are there with all the primes below 1 million?

144

What's the longest run of arithmetically sequenced primes below 1 million?

The longest run is 15 primes

DaveE
 

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