question about parallel sequence making prime
Brendan McKay
bdm at cs.anu.edu.au
Wed Feb 21 14:19:38 CET 2007
Hi, This problem occurred in my formal languages course.
I'm guessing there is some interesting exploration possible.
Consider a pair of positive integer sequences
a(1),a(2),a(3),...
b(1),b(2),b(3),...
with this property:
* for all i, a(i)+b(i) is prime
* for all i,j not equal, a(i)+b(j) is composite
Example:
1 3 7 8 24 25 27 32 48 ...
1 8 24 3 7 48 32 27 25 ...
Thus 24+7 is prime, but 24+1,24+8,24+24,24+3,24+48,24+32,24+27,24+25 aren't.
It is fairly clear that such a sequence exists. Choose in the
order a(1),b(1),a(2),b(2),... and probably <I didn't do it!> it
is not hard to prove that a choice is always available.
I made the example in this fashion starting with a(1)=1 and
always choosing the smallest available value. For example,
b(3)=24 is the smallest number such that a(1)+b(3) and
a(2)+b(3) are composite but a(3)+b(3) is prime.
Note the curious way that the second sequence looks like a
permutation of the first (so far). Why is that?
Brendan.
If the terms of the sequences are chosen greedily in the order a(1),
b(1), a(2), b(2), etc. then the a- sequence has to be increasing.
Given that, since b(10) = 504 and a(25) = 468, a(26) = 519, the two
But the rate of growth of the sequences, as well as what percentage of the
b-sequence terms are contained in the a-sequence, is an interesting
question.
Jeffrey Shallit
> From seqfan-owner at ext.jussieu.fr Wed Feb 21 08:21:18 2007
> From: Brendan McKay <bdm at cs.anu.edu.au>
> To: seqfan at ext.jussieu.fr
> Subject: question about parallel sequence making prime
>
> Hi, This problem occurred in my formal languages course.
> I'm guessing there is some interesting exploration possible.
>
> Consider a pair of positive integer sequences
> a(1),a(2),a(3),...
> b(1),b(2),b(3),...
> with this property:
> * for all i, a(i)+b(i) is prime
> * for all i,j not equal, a(i)+b(j) is composite
>
> Example:
> 1 3 7 8 24 25 27 32 48 ...
> 1 8 24 3 7 48 32 27 25 ...
> Thus 24+7 is prime, but 24+1,24+8,24+24,24+3,24+48,24+32,24+27,24+25 aren't.
>
> It is fairly clear that such a sequence exists. Choose in the
> order a(1),b(1),a(2),b(2),... and probably <I didn't do it!> it
> is not hard to prove that a choice is always available.
> I made the example in this fashion starting with a(1)=1 and
> always choosing the smallest available value. For example,
> b(3)=24 is the smallest number such that a(1)+b(3) and
> a(2)+b(3) are composite but a(3)+b(3) is prime.
>
> Note the curious way that the second sequence looks like a
> permutation of the first (so far). Why is that?
>
> Brendan.
>
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