A Question to Trouble Even the Best of You

adityab88

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.

Gib Z

There is no such sequence.

Moo Of Doom

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.

Gib Z

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.

arildno

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.

christianjb

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

(Actually, it sounds more like a Google problem)

Gib Z

lol adityab88 you google recruit!!!!

Rhythmer

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

Gib Z

No idea don't ask me lol.

If you want, start a thread here inquiring about this. But not C++ I know.

Rhythmer

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

d_leet

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..

Rhythmer

For all terms in the sequence:

[tex] X_n = X_{n-1} + ( (n-1) * 10 ) [/tex]

Starting from (n = 1), [tex] X_0 = 1 [/tex]



Is my solution too bad?

d_leet

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

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

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

adityab88

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!

Data

The spacing also can't be 30:

7, 37, ..., 187 doesn't work because 187 = 11*17. But for any prime [itex]p \neq 7[/itex], we have [itex] p \equiv 2k[/itex] (mod 7) for some k, [itex]1 \leq k \leq 6[/itex]. 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

Gib Z

Thats top stuff there Data.

adityab88

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.