Ten prime numbers describing an arithmetic sequence

[K Sengupta]

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?

 

[disregardthat]

wrong

 

[Rogerio]

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 ?

 

[Jimmy Snyder]

Spoiler

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?

 

[davee123]

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

 

[Georgepowell]

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

 

[Jimmy Snyder]

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.

 

[davee123]

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

 

[davee123]

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?

Spoiler

53813

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

Spoiler

144

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

Spoiler

The longest run is 15 primes

DaveE