Problems & Puzzles: Puzzles

Puzzle 626. Frank Rubin, kissing primes

One more time and after the generous acceptance of my friend Frank Rubin, I bring -from one of his always interesting pages in his web site Contest Center- the following nice puzzle.

This time the issue will be around his four puzzles devoted to "kissing primes"

He defines a set of primes as a set of kissing primes if every pairwise concatenation of them produce a new prime.

Of course, if the set of kissing primes has K members, the number of produced primes will be K(K-1)

Let K=3, and the set of kissing primes to be {A, B, C} then the produced primes are: AB, BA, AC, CA, BC, CB

An example computed by me:
For K=6, {3, 7, 109, 673, 129976621, 268976437}

Q1. Send the smallest set of kissing primes for the following K values: 2, 3, 4, 5, 6, 7,  8, ...,? Smallest set means here smallest sum of the K primes of the set.

As you can see in the Rubin's pages, it's supposed that there is a limit of the K possible value for this kind of sets.

Q2. What can you comment about this limit? Exist? What is your computed value?

Contributions came from J. K. Andersen & Hakan Summakoglu


Jens wrote:

K=2: {3, 7} sum=10
K=3: {3, 37, 67} sum=107
K=4: {3, 7, 109, 673} sum=792
K=5: {13, 5197, 5701, 6733, 8389} sum=26033
K=6: {25819, 29569, 209623, 234781, 422089, 452041} sum=1373922


A non-minimal case for K=8: {3, 7, 109, 673, 20734460559829,
33218902471771, 56960675763007, 763515726369961} sum=874429765165360


I haven't found a reason for a limit. If there is no specific reason
such as a pattern in factorizations then heuristics point to no limit.


Hakan wrote:

My smallest kissing primes are: 
For K=2, {3, 7}
For K=3, {3,37,67}
For K=4, {3,7,109,673}
For K=5, {13,5197,5701,6733,8389}
For K=6, {3,7,229,347161,6971863,8307631} [Not minimal, against the Jens solution]


Frank Rubin wrote on March 3, 2012:

A set of 8 is very impressive.  As I said, the known limit, proved by Lee Morgenstern, is quite high.




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