Re: [PrimeNumbers] PrimePuzzles.net
- Jose Ramón Brox wrote:
> I just clicked http://www.primepuzzles.net/ and I didn't get any problems.Sure you did not get your own cache?
If you are sure: I know little about the Internet. Could it be an error on a
regional domain name server or whatever it's called. I am trying to access
from Denmark. Can you e-mail Carlos and tell him about the problem others are
Jud McCranie wrote:
> I can't get onto PrimePuzzles.net (last night or this morning). It takesme to
> http://www.landois.com/ instead. If I try to email Carlos Rivera, theI cannot access www.primepuzzles.net either. I also get
> message is rejected. Are others having the same problem? Anyone know
> what's going on?
I get a second error for www.primepuzzles.net/puzzles (browser switches
between two messages in status line) and a third for
www.primepuzzles.net/puzzles/puzz_209.htm (nothing happens).
If anyone is interested in the solution for puzzle 209 which should have come
today, my mail to Carlos is below. I don't know what the new puzzle or problem
today should be.
--- Begin e-mail to Carlos Rivera ---
Puzzle 209. Triangles of primes
1. Can you provide a formula to calculate the quantity of embedded equilateral
triangles in an K-triangular array?
This has to be in the EIS. A quick lookup for the first terms 1,5,13,27 finds
ID Number: A002717 (Formerly M3827 and N1569)
Comments: Number of triangles in triangular matchstick arrangement of side n.
Who needs to think when the EIS is there :-)
2. Can you find one solution for every 4<K<=10?
Each solution can be rotated and mirrored in 6 ways. I only count this as one
There is a single solution for K=3.
There are 104 solutions for K=4.
There are 1261 solutions for K=5. This is one of them:
7 29 5
23 17 13 19
53 3 11 47 43
There are no solutions for K>5.
I first tried my search program for K=6 and found no solutions. I suspected
there would never be more solutions. This can be proved by considering
potentiel solutions modulo 3. The only prime which is 0 (modulo 3) is 3. The
sum of 3 triangle vertexes must not be divisible by 3 if it has to be a prime.
This means the modulo 3 triples (0,1,2), (1,1,1) and (2,2,2) are impossible.
I quickly modified my existing search program to show there are no solutions
for K>5, not even if the 0 (from 3) is omitted. This could also be proved on
paper relatively easy.
The above means there are no K>5 solutions for any set of distinct primes.
There are solutions for K=6 if we allow repetitions of 3. This modulo 3
template has two 0's, corresponding to two 3's:
2 0 2
1 2 2 1
2 1 1 1 2
1 1 2 2 1 1
There is only one other modulo 3 template with only two 3's. That is the above
with all 1's and 2's swapped. This would give one 2 too few to match the
smallest primes modulo 3.
If we replace the largest prime 73 with a second 3 then we get 214 solutions,
all matching the above template. Here is one of them:
11 3 23
13 29 41 37
59 31 67 19 47
61 7 5 17 43 73
For K>6 there are no template solutions and thus no prime solutions (or even
integer solutions with prime sums) for any number of 3's allowed.
3. Do you devise a systematic approach in order to get the solutions asked in
I used pretty brute force for K<=6. My C program went through the vertices
with a recursive function. It placed each allowable prime at a vertex before a
recursive call for the next vertex. It seemed like it would be too slow for
K=7 and I did not expect solutions. I thought of the modulo 3 simplification
instead of optimizing the program.
--- End e-mail to Carlos Rivera ---
Jens Kruse Andersen