# [seqfan] Re: An interesting problem

Lang, Wolfdieter (ITP) wolfdieter.lang at partner.kit.edu
Thu Jul 6 11:18:42 CEST 2023

```Dear Gary,

I am not sure that I can look into this right now. I am prepareing a paper on Cantor's real  algebraic numbers.

Don't you want to add your original sequemce of the old name  to OEIS by acting with the inverse Pascal triangle (signed old one) on the sequence of the new name in A135042?

Best regards,

Wolfdieter

________________________________
Von: SeqFan <seqfan-bounces at list.seqfan.eu> im Auftrag von Max Alekseyev <maxale at gmail.com>
Gesendet: Mittwoch, 5. Juli 2023 21:56:14
An: Sequence Fanatics Discussion list
Betreff: [seqfan] Re: An interesting problem

A possible generalization is going in 2D. Here are 6x6 and 8x8 squares
filled with numbers 1 to 6^2 and 1 to 8^2 respectively such that the sum of
numbers in any two adjacent cells (sharing a side) is prime. Furthermore,
this property holds cyclically, ie. these squares can be considered on a
torus.

|35|18|13|24|29|02|
|36|01|16|07|30|17|
|25|22|21|10|31|12|
|34|09|20|03|28|19|
|33|08|23|14|15|04|
|26|11|06|05|32|27|

|08|45|22|15|58|09|14|05|
|33|28|61|52|01|10|57|26|
|46|25|36|31|16|07|04|63|
|13|06|11|12|55|34|39|40|
|30|17|56|41|42|37|64|43|
|59|20|27|02|29|60|19|24|
|54|47|32|21|50|53|48|49|
|35|62|51|38|03|44|23|18|

Regards,
Max

On Tue, Jul 4, 2023 at 5:31 AM Brendan McKay via SeqFan <
seqfan at list.seqfan.eu> wrote:

> https://mathoverflow.net/questions/241569
>
> Note that the question asks for a path, while most of the answers
> concern a cycle (where the first and last elements must also add
> to a prime).
>
> This doesn't matter. For a cycle note that n must be even; in that case
> just cut any edge if you only want a path.  If n is odd, add n+1 to the
> mix, find a cycle, then delete n+1 to get a path.
>
> If I understand it, Johan Wästlund's answer on that page proves it
> up to 10^13.
>
> A051252 and A228917 are mentioned and relevant.
>
> A version which I haven't seen anywhere is whether you can find a
> path with specified endpoints.  I'll spell it out:  For n >= 2,
> and distinct a,b in {1..n}, is there a permutation of 1..n that starts
> with a, ends with b, and consecutive elements sum to a prime?
> There are trivial parity constraints (exercise for the reader):
>     If n is odd, a and b are both odd.
>     If n is even, a and b have opposite parity.
>
> For tiny n there are some cases that can't be done.
> These are impossible:  n=5, {a,b}={1,3} and n=6, {a,b}={1,2}.
>
> For 7 <= n <= 750, all cases work if they meet the parity constraints.
> It seems reasonable to conjecture that this continues forever.
>
> Here are solutions for n=7 and all {a,b}:
>
>   1 4 7 6 5 2 3
>   1 2 3 4 7 6 5
>   1 4 3 2 5 6 7
>   3 2 1 4 7 6 5
>   3 4 1 2 5 6 7
>   5 2 3 4 1 6 7
>
> I checked that  {a,b}={1,n} works for 2 <= n <= 33,000.
> I also checked that {a,b}={1,2} works for even 8 <= n <= 20,000 .
>
> Brendan.
>
> On 2/7/2023 10:44 pm, Yifan Xie wrote:
> > Hi,
> > I recently discovered a problem:
> > For an integer n>=2, there exists a sequence {a(n)} consisting of 1, 2,
> 3, ... , n that for all 1<=i<=n, the sum of a(i) and a(i+1) is a prime.
> > Do all integers n>=2 satisfy the above condition?
> > The easiest algorithm to find a possible sequence is that all terms are
> the largest possible ones. For example, for n=5, the sequence starts with
> 5, since 5+4 and 5+3 are not primes, the next term is 2. Similarly, thw
> whole sequence is {5, 2, 3, 4, 1}.
> > I tested this algorithm for n<=10^4 and found that only 2108 and 7288
> failed.
> > Can anyone help?
> >
> >
> > Cheers,
> > Yifan Xie (xieyifan4013 at 163.com)
> >
> > --
> > Seqfan Mailing list -http://list.seqfan.eu/
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

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