Problems & Puzzles: Puzzles

Puzzle 367. Two dice to produce prime numbers Jaime Ayala proposes the following puzzle: Design the set of (12) numbers on the faces of a couple of (6 faces per die) dice in order that the sum of the numbers in every possible (36 possibilities) pair of faces - when the dice are thrown - produces a prime number. The design unique restriction is that you can not repeat the numbers assigned to the faces of the dice. Questions. 1. Make a design with the minimal sum of the 12 numbers assigned to the 12 faces. 2. Redo the design if the 36 primes produced must be distinct primes.

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Complete & correct solutions to both questions came from Daniele Degiorgi & Fred Schneider. Partial solution came from Igor Schein, Stuart Gascoigne & W. Edwin Clark. Approximations to the minimal solutions were sent by Anurag Sahay. Stuart & Clark make other kind of interesting contributions to this puzzle.

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From Daniele & Fred:

There are two minimal solutions for question 1, all summing to 384

{0,6,12,26,56,62} {5,11,17,41,47,101}
{0,6,12,36,42,96} {5,11,17,31,61,67}

and there are two minimal solutions for question 2 (sum is 774)

{0,6,30,96,126,210} {13,31,41,53,67,101}
{0,18,28,40,54,88} {13,19,43,109,139,223}

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Stuart wrote:

An easy (non-mimimal) answer to Q1 can be obtained from a sequence of 11 primes
in arithmetic progression eg 23143 + 30030k (k=0 to 11)
Die 1 = 23143, 23143+30030*1, 23143+30030*2, 23143+30030*3, 23143+30030*4,
23143+30030*5
Die 2 = 0, 30030*1, 30030*2, 30030*3, 30030*4, 30030*5.

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Edwin wrote (bilingual set of contributions):

For each positive integer n define f(n) = greatest positive integer m such that there exists sets of non-negative integers A and B such that
     (i)  A has n elements.
     (ii) B has m elements
     (iii) For every a in A and b in B the sum a + b is prime.
And set f(n) = infinity, if for some A of size n the set B can be chose to be infinite

Partial Results:
 f(1) = infinity  (Since by Dirichlet's Theorem for any prime p there are infinitely many primes of the form p + k).

Using Maple I can show:
   f(2) >536
   f(3) > 239
   f(4) > 121
   f(5) > 63
   f(6) > 35
   f(7) > 25
For these results I found A among the 45 odd primes less than 200 and for B among the even numbers from 0 to 10,000.
I'm sure someone using something like C could do much better.

A number of questions arise here:
(a) Is it possible that one can find infinite sets A and B of non-negative integers such that for all a in A and for all b in B we have a + b is prime. (It is easy to see that without loss of generality we may assume A contains only odd primes and B contains only even numbers and  0. )
(b) If (a) is false is it true that for all n we have f(n) >= n?
(c) What happens if we look for k > 2 sets A1, A2, ...,Ak of non-negative integers such that for every a1 in A1, a2 in A2,..., ak in Ak the sum a1+a2+...+ak is prime.
 

Remark: We can view this as a problem in graph theory. Let G be the infinite bipartite graph with parts X = all odd primes , Y = non-negative even integers where x in X and y in Y are adjacent when x + y is prime. Then we seek subgraphs of G which are complete bipartite graphs.

...

Acabo de encontrar un diseño para 3 "dados" de 3 caras que siempre dan
primos cuando se tiren:

A={2, 18, 32},
B={0, 24, 30},
C={5, 11, 41}.

Es decir, para todos x en A, y en B, z en C , x + y + z es primo.

...

Andrew Granville proved UNDER THE ASSUMPTION of the prime k-tuples
conjecture that (among other things) the following hold:

1. There are infinite sets of positive integers A and B such that for
all a in A and b in B, the sum a + b, is prime..

2. If N is any odd number there are N infinite sets A1, A2,...,AN of
distinct primes such that for all a1 in A1, a2 in A2,...,aN in AN, the
sum a1 + a2 + ...+aN is prime.

The paper by Andrew Granville: "A note on sums of primes", Canad. Math.
Bull. 33 (1990), no. 4, 452--454, can be found online at
http://www.dms.umontreal.ca/~andrew/PDF/sumprime.pdf

...

Great idea! Using Gascoigne's idea one can use
the longest known arithmetic progression of primes
which is of length 22 to find 3 dice (each of 6 faces)
so that the sum when the three dice are thrown
is always prime. The only problem is that the
dice would have to be large to hold the numbers.

Also supposedly it has been proved that there
are arbitrarily long arithmetic progressions of
prime:

Green, B. and Tao, T. "The Primes Contain Arbitrarily Long Arithmetic
Progressions." arXiv:math.NT/0404188 v1 preprint. Apr. 8, 2004.
http://arxiv.org/abs/math.NT/0404188.

From this one can obtain finite sets A1, ..., An of
any desired size such that the set of sums A1 + ... + An
contains only primes. Well, this settles existence,
but finding them is another matter!

De todos modos es un problema muy interesante!

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This means that having 36 primes in AP: p, p+k, p+2k, ... , p+35k, we can get a non-minimal solution to Q2 as this one:

{p, p+k, p+2k, p+3k, p+4k, p+5k} + {0, 6k, 12k, 18k, 24k, 30k}

The good new is that we know that such thing as 36 primes in AP exists. The bad new is that nowadays (2006) we don't know one single specific example.

Hopefully soon (when?) we will know one example of 36 primes in AP.

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J. K. Andersen wrote:

Jörg Waldvogel and Peter Leikauf currently list 77 occurrences of a pattern
with 17 primes in
http://www.sam.math.ethz.ch/~waldvoge/Projects/clprimes05.pdf
This gives two sets with 17 and 77 numbers where each sum is a distinct
prime.
I used their data in a small search which found an 18th number that gives
a prime sum with 27 of the 77 numbers.
This gives sets with 18 and 27 numbers having distinct prime sums.
Set 1: {0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68,
9126650}
Set 2: {71542018620258822095351, 127372818047327460718001,
472773823866578374260011, 668134176357714775428611,
842902600507150788234521, 1448301799940226082771811,
2638263731291524361295161, 2965402830695807653891631,
4184454523805320162938431, 5188248562214464094307851,
6462943397426544261913571, 6515331587478752597979311,
6732639026163593595722561, 7068848143689559074537581,
7106614823259270051322481, 8394208478031243473535821,
8796828522725933171551961, 8941097330542117721700611,
9867358577329746770637251, 9980642183707047492649901,
10508599120346583663997481, 11836184125186489787548721,
11985312529091443124703011, 12314036812755277924784351,
13114239319848333922324961, 15307029480404445010527191,
15722057448876967802383661}

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