# Ten prime numbers describing an arithmetic sequence

• K Sengupta
In summary, the ten prime numbers, each less than 3000, when arranged in increasing order of magnitude describe an arithmetic sequence. The sum of the prime numbers p and p+d are even, but also equal to the prime number p+3d, a contradiction. However, p+p+d = p+3d because p+p+d is the sum of the prime numbers p, p+2d, and p+d. This is a simple mistake. There are more runs of 9 besides the 2 included within your run of 10. The longest run of arithmetically sequenced primes below 1 million is 8.
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?

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?

jimmysnyder said:

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

## What is an arithmetic sequence?

An arithmetic sequence is a series of numbers in which the difference between any two consecutive terms is constant. For example, the sequence 2, 5, 8, 11, 14 is an arithmetic sequence with a common difference of 3.

## What are prime numbers?

Prime numbers are numbers that are only divisible by 1 and themselves. They cannot be divided evenly by any other numbers.

## How do you find ten prime numbers in an arithmetic sequence?

To find ten prime numbers in an arithmetic sequence, you can start with any prime number and add the common difference repeatedly until you have ten prime numbers. For example, if you start with the prime number 5 and a common difference of 3, you would get the sequence 5, 8, 11, 14, 17, 20, 23, 26, 29, 32.

## What is the importance of ten prime numbers in an arithmetic sequence?

Ten prime numbers in an arithmetic sequence can be useful in mathematical patterns and equations. They can also provide insights into the distribution and relationships of prime numbers.

## Can ten prime numbers in an arithmetic sequence be found in real-world applications?

Yes, ten prime numbers in an arithmetic sequence can be found in real-world applications such as cryptography, where prime numbers are used in creating secure codes and algorithms. They can also be found in nature, such as the pattern of leaf arrangement on a stem or the spiral arrangement of seeds in a sunflower.

• Calculus and Beyond Homework Help
Replies
22
Views
3K
• General Math
Replies
35
Views
2K
• General Math
Replies
8
Views
492
• General Discussion
Replies
4
Views
2K
• Precalculus Mathematics Homework Help
Replies
9
Views
1K
• Engineering and Comp Sci Homework Help
Replies
32
Views
3K
Replies
0
Views
546
• Engineering and Comp Sci Homework Help
Replies
1
Views
559
• Math Proof Training and Practice
Replies
1
Views
1K
• General Math
Replies
1
Views
1K