# Prime Number Arithmetic Progression

• FeDeX_LaTeX
In summary: , in summary, the least possible value of the largest term in an arithmetic progression of seven distinct primes is 97.
FeDeX_LaTeX
Gold Member
"Determine the least possible value of the largest term in an arithmetic progression of seven distinct primes."

I really have no clue what to do here. Is there a general tactic that you can use to do this, other than trial and error? Some experimenting gives you these of arithmetic progressions:

5, 11, 17, 23, 29
5, 17, 29, 41, 53
7, 19, 31, 43
3, 7, 11
41, 47, 53, 59
61, 67, 73, 79
7, 37, 67, 97, 127, 157
107, 137, 167, 197, 227, 257
53, 113, 173, 233, 293, 353

I haven't found one that gives me a string of 7 primes yet and I've just been looking at primes under 100.

Of course there are general rules to follow when finding the strings that I spotted (shouldn't add a number to a prime which will land you on a multiple of 5, such as adding 12 to a prime excluding 2).

EDIT: Okay, I think I know how to solve this problem. If I had 4, 6 or 8, I'll never get a streak longer than 5, but if I choose a difference of +10 (or a multiple of 10), I might find one more easily.

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Remember, Prime Numbers except 2 or 3 can be expressed as 6k±1.

AGNuke said:
Remember, Prime Numbers except 2 or 3 can be expressed as 6k±1.

Thanks for this. So would it be worthwhile to only consider the primes that are 1 and 5 (mod 6)?

I assume that it would better help you to determine the prime number and thus the relevant progression.

Start from k=1, we get 5 and 7. Both are Primes.
k=2, we get 11 and 13.
k=3, we get 17 and 19.
k=4, we get 23 and 25, not a prime. And so on...

It may help you to get a proper listing, like it is probable that AP can be formed with common difference of 6, 12...

You can also look for Prime Generating Programs, if you want I can give you one.

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But I thought that the required arithmetic progression here had to have a common difference that was a multiple of 10? All primes (except 2) are odd, and if you add a common difference that is a multiple of 6 to an odd prime, you will quickly end up with a multiple 5 and you will be unable to get an AP longer than 5 terms.

If the difference is 10 which is 1 mod 3 you get a number divisible by 3 at every third step. So you need to include 3 into the difference, it must be at least 2*3*5=30. That is 3 mod 7 and you will bump into a multiple of 7 in every 7th step...

ehild

## What is a Prime Number Arithmetic Progression?

A Prime Number Arithmetic Progression is a sequence of prime numbers in which the difference between any two consecutive numbers is constant. For example, 3, 7, 11, 15 is a Prime Number Arithmetic Progression with a common difference of 4.

## How do you find the next number in a Prime Number Arithmetic Progression?

To find the next number in a Prime Number Arithmetic Progression, you can add the common difference to the previous number in the sequence. For example, if the previous number is 11 and the common difference is 4, the next number would be 15.

## What is the formula for the nth term in a Prime Number Arithmetic Progression?

The formula for the nth term in a Prime Number Arithmetic Progression is n x d + a, where n is the term number, d is the common difference, and a is the first term in the sequence. For example, the 5th term in the sequence 3, 7, 11, 15 would be 5 x 4 + 3 = 23.

## What is the significance of Prime Number Arithmetic Progressions in mathematics?

Prime Number Arithmetic Progressions have been studied for centuries and have many applications in mathematics, including in number theory, algebra, and cryptography. They also help us better understand the distribution of prime numbers and their properties.

## Are there any famous unsolved problems related to Prime Number Arithmetic Progressions?

Yes, there are several famous unsolved problems related to Prime Number Arithmetic Progressions, including the Green-Tao theorem, which states that there are arbitrarily long arithmetic progressions of prime numbers, and the Twin Primes Conjecture, which suggests that there are infinitely many pairs of prime numbers that differ by 2.

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