# Prime Number Arithmetic Progression

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

Gold Member
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)?

AGNuke
Gold Member
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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Gold Member
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.

ehild
Homework Helper
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