# A Question to Trouble Even the Best of You

DAn arithmetic sequence is a sequence of numbers where the difference between consecutive terms is a constant. For example, 2, 5, 8, 11 is an arithmetic sequence with a common difference of 3. The formula to find the nth term of an arithmetic sequence is: an=a1+(n-1)d, where a1 is the first term and d is the common difference. In the conversation above, the problem was to find the smallest possible arithmetic sequence consisting of seven primes.

Find the smallest possible arithmetic sequence consisting of seven primes.

For example, the smallest possible arithmetic sequence consisting of five primes is: 5, 11, 17, 23, 29.

By the way, by "smallest possible" I mean that the last term in the sequence must be the smallest possible of all such sequences.

There is no such sequence.

Gib Z, you might want to think about that statement some more. It has been proven that there are arithmetic sequences of primes of any finite length.

Shhh! lol

Ill reword it then, There is no such sequence that I can be bothered to find and that has any mathematical value. To find such a sequence all one needs is some programming knowledge and to be really bored.

What can readily be established is that any member of such a sequence must have the same last digit.

So there, I have narrowed it down immensely.

Yeah, is this a computing problem or a math problem?

(Actually, it sounds more like a Google problem)

Gib Z said:
To find such a sequence all one needs is some programming knowledge and to be really bored.

What type of programming do we need to solve this sort of mathematical problems? C++ ?

No idea don't ask me lol.

Gib Z said:
No idea don't ask me lol.

Unfortunately, C++ is the only programming language I can use.

Regarding adityab88's question, here is a solution I've just gotten:
1, 11, 31, 61, 101, 151, 211, 281
And it's the smallest possible because 361 is not a prime

Rhythmer said:
Unfortunately, C++ is the only programming language I can use.

Regarding adityab88's question, here is a solution I've just gotten:
1, 11, 31, 61, 101, 151, 211, 281
And it's the smallest possible because 361 is not a prime

However this is not an arithmetic sequence because consecutive terms do not differ by one single constant.

ie. 11-1=10, whereas 31-11=20, 61-31=30 etc..

d_leet said:
However this is not an arithmetic sequence because consecutive terms do not differ by one single constant.

ie. 11-1=10, whereas 31-11=20, 61-31=30 etc..
For all terms in the sequence:

$$X_n = X_{n-1} + ( (n-1) * 10 )$$

Starting from (n = 1), $$X_0 = 1$$

:uhh:

Rhythmer said:
For all terms in the sequence:

$$X_n = X_{n-1} + ( (n-1) * 10 )$$

Starting from (n = 1), $$X_0 = 1$$

:uhh:

The only thing wrong with it is that it is not an arithmetic sequence.

An arithmetic sequence has terms of the form: an=a0+n*d

Where a0 is the first term, and d is the common difference between terms.

hello ... ppl
i am quite sure you do not need programming equipment.
just your brains will suffice; i am sure of this becuase i saw this question on a maths paper, where computers were not allowed

this is all i have: the difference between the consecutive primes has to be a multiple of 30, i have shown this using some number theory. so, for example, the difference could be 60; a sequence like this (7, 67, 127, ...).

good luck and have fun!

The spacing also can't be 30:

7, 37, ..., 187 doesn't work because 187 = 11*17. But for any prime $p \neq 7$, we have $p \equiv 2k$ (mod 7) for some k, $1 \leq k \leq 6$. Since 30 mod 7 = 2 that means that p+30q is divisible by 7 for some q, 1<q<=6.

That can be generalized to remove some other spacings too; I'm tired and will figure it out in the morning!

Edit: OK, I lied about figuring it out later!

You can also eliminate all other spacings, except those which are products of 7, in cases for any sequence which does not start with 7. So the sequence has to start with 7, or the spacing is a multiple of 7 (ie. 210, 420, etc.).

A little bit of experimenting then yields the sequence: 7, 157, 307, 457, 607, 757, 907

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Thats top stuff there Data.

good stuff

Data said:
The spacing also can't be 30:

7, 37, ..., 187 doesn't work because 187 = 11*17. But for any prime $p \neq 7$, we have $p \equiv 2k$ (mod 7) for some k, $1 \leq k \leq 6$. Since 30 mod 7 = 2 that means that p+30q is divisible by 7 for some q, 1<q<=6.

That can be generalized to remove some other spacings too; I'm tired and will figure it out in the morning!

Edit: OK, I lied about figuring it out later!

You can also eliminate all other spacings, except those which are products of 7, in cases for any sequence which does not start with 7. So the sequence has to start with 7, or the spacing is a multiple of 7 (ie. 210, 420, etc.).

A little bit of experimenting then yields the sequence: 7, 157, 307, 457, 607, 757, 907

good stuff
how long did it take you to "experiment" and find the final sequence? cause i was looking for it, too, but my experimentation did not really work.

About five seconds :tongue:. It makes sense to start at 7; if you can find a sequence starting with 7 that has spacing smaller than 210 (and there's no smaller sequence starting with 7), then that's obviously the smallest one.

Of course, from what I've done there was no particular reason to confine myself to sequences starting with 7, so it would be fair to consider it a lucky coincidence.

What is an "arithmetical sequence"?

An arithmetic sequence is one of the form $(n,n+k,n+2k,n+3k,...,n+mk,...)$.

Data said:
About five seconds :tongue:. It makes sense to start at 7; if you can find a sequence starting with 7 that has spacing smaller than 210 (and there's no smaller sequence starting with 7), then that's obviously the smallest one.
Ah,yes,yes,...
To me it makes sense to start with 199 and find "minimum" prime-sequence of 10 terms with spacing 210 :

$$199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089$$.

But it took me longer time ,about 10 seconds:tongue:

DaveC426913 said:
What is an "arithmetical sequence"?

I've written a mini-introduction to arithmetic progressions https://www.physicsforums.com/blogs/edgardo-22482/arithmetic-progressions-mini-introduction-887/ [Broken]

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are there any non-primes in there?

but even if that is right (i doubt it is, I'm too tired to see the factors i probably should) i cheated b/c i guessed

mr200backstrok said:

are there any non-primes in there?

but even if that is right (i doubt it is, I'm too tired to see the factors i probably should) i cheated b/c i guessed

51 and 81 are both composite.

oh

haha 9*9 = 81 yep I am tired

Any idea where i could get a description of the concepts they are talking about? (I don't want to hijack the thread)

mr200backstrok said:
Any idea where i could get a description of the concepts they are talking about? (I don't want to hijack the thread)

All I used for this problem was a little bit of modular arithmetic.

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