Problems & Puzzles: Puzzles

 Puzzle 773. Necklaces with k primes  Jean Brette sent the following nice puzzle: Let  S(k) = ( p1, p2, , pk)  be a sequence of  k  odd distinct primes, and  N(k) its associate sequence  (p1, , pk, p(k+1), p(k+2)), where  p(k+1) = p1 and p(k+2) = p2.   We can see the sequence   S(k)   as a circular necklace of  k  prime pearls, and we can define a new necklace  S(k) = (q1, q2, qi, .., qk) where each pearl  qi  is the sum of three consecutive pearls of  N(k) :  qi = pi + p(i+1) +p(i+2).  (1 ≤ i ≤ k)   If all the  qis   of   S(k)  are primes, I call   S(k) a  good necklace.      (G) If all the  qis  are distinct primes, I call  S(k)  a very good necklace   (VG)   Examples :   k = 3 :  S (3) = (3,    5,     7)    =>   S(3) = (15,   15,  15)          is not a good necklace          :  S (3) = (3,    5,    11)   =>   S(3) = (19,   19,  19)          is a good necklace     G   k = 4 :  S(4) = (3,  5, 11,  7 )   =>    S(4) = (19, 23, 21, 15) .    is not good             S(4) = (7, 11, 13, 23)  =>    S(4) = (31, 47, 43, 41).     is very good       VG          k = 7 :  S(7) = (3, 5, 11, 7, 13, 17, 23)  =>  S(7) = (19, 23, 31, 37, 53, 43, 31) is only  G   In order to compare two sequences of the same length   k  , we need a criterion.  I choose the following :   For a sequence  S, let be  P(S)  the product of all the terms of  S and its S' :   P(S) = p1* p2*...*pk * q1*q2*....* qk. To avoid very big numbers, let be  I(S) = log (P(S)) .   A sequence S1(k) is said better than an other sequence  S2(k)   if   I(S1) <  I(S2) .   Examples :  When  k = 6,  the three follwing sequences  S1, S2, S3 are VG S1 =  5, 17,  7, 13, 11, 37    =>   S'1 = 29, 37, 31, 61, 53, 59            VG      I(S1) = 14,96           S2 =  7, 13, 11, 19, 37, 17   =>   S'2 = 31, 43, 67, 73, 61, 37            VG      I(S2)  = 16,3   S3  = 3,  7, 13, 11, 19, 31    =>    S'3 = 23, 31, 43, 61, 53, 41           VG      I(S3)  = 14, 38   So  S3 is better than S1, which is better than S2.     Q1 : Does it exist a good  (or a very good) sequence S(k) for each  number  k ? Q2 : For each k, could you give your own records (i.e. your S(k) with the smallest   I(S(k))  ) Q3 : Iteration : does it exist  some very good sequence  S(k) such that  S(k) is  also good (or very good) ? etc.

Contributions came from Jan van Delden, Emmanuel Vantieghem and Jean Brette

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Jan wrote:

Q1: Yes k>=3, no proof.

Q2:  Solutions which are VG with smallest index I

k  I         Necklace
4  10.25  [19, 5, 7, 17]
5  12.77  [19, 11, 7, 5, 17]
6  14.63  [23, 3, 11, 5, 7, 17]
7  18.25  [31, 13, 3, 7, 19, 5, 17]
8  21.28  [41, 7, 11, 5, 3, 23, 17, 13]
9  24.26  [41, 13, 17, 23, 3, 5, 11, 7, 19]
10 27.65 [41, 19, 7, 5, 11, 3, 23, 17, 13, 29]
11 31.10 [41, 19, 7, 5, 11, 3, 23, 17, 13, 31, 29]
12 34.76 [41, 19, 7, 5, 11, 3, 23, 17, 31, 13, 29, 37] *
13 38.64 [47, 13, 23, 17, 3, 11, 5, 7, 29, 31, 41, 37, 19]
14 42.43 [47, 13, 29, 31, 37, 41, 23, 3, 5, 11, 7, 19, 17, 43] *
15 46.46 [53, 31, 29, 37, 43, 47, 41, 13, 17, 23, 3, 5, 11, 7, 19] *
16 50.45 [59, 31, 19, 47, 41, 43, 29, 7, 5, 11, 3, 17, 23, 13, 53, 37] *
17 54.44 [61, 37, 29, 43, 41, 19, 7, 5, 11, 3, 23, 17, 13, 53, 31, 47, 59] *

* Indicates that the largest prime in necklace S(k) is equal to prime(k+1), for these solutions its not necessary to check a larger prime in the necklace for smaller index I.
However they contain the prime 3 and hence there will be no extra iterate (check mod 6).

Q3: There are no necklaces with odd length and iterates larger than 1. The proof also uses addition mod 6.

Given a necklace with primes p[1]..p[k], put these in a vector p[0], our starting necklace.
The next one p[1] can be found by finding a suitable matrix A: p[1]=A.p[0].
For our iterates we have: p[n]=A^n.p[0].

If k=4 this behaves particularly nice. We can write A = J  C, with the 4x4 matrix, all elements equal to 1 and
C equal to [e[2],e[3],e[4],e[1]], with e[j] the elementary columnvectors. C^j behaves cyclic: C^4=I, the identity matrix.

Skipping some algebra, we find: A^4 = I+20J.
Or if we compute the effect of A^4 on a certain element of p[i]: p[i,j]:
p[i+4,j]=(A^4 p[i])[j] = p[i,j] + 20* sum(p[i,j],j=1..4)
Which tells us that we only have to compute A,A^2,A^3, the rest follows by adding sums of the primes.
Furthermore these sums increase by a factor 3 in each step, so only sum(p[0,j],j=1..4) has to be computed.

If one wants to find a necklace having at least n>=4 we could reduce our combinations of p[0,j] by directly checking if the right hand side is prime as well. We only have to check an ordered sequence of primes p[0,j].

The necklace with smallest index 20.01 and 4 iterates:
[[673, 17, 53, 397], [743, 467, 1123, 1087], [2333, 2677, 2953, 2297], [7963, 7927, 7583, 7307], [23473, 22817, 22853, 23197]]

The first necklace with 5 iterates, index=25.39:
[[3001, 257, 271, 1091], [3529, 1619, 4363, 4349], [9511, 10331, 12241, 9497], [32083, 32069, 31249, 29339], [95401, 92657, 92671, 93491], [280729, 278819, 281563, 281549]

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Emmanuel wrote:

My best solution for  S(20) :
{3,5,11,7,13,17,23,19,31,29,47,37,43,59,61,71,67,73,41,53}  with  I(S(20)) = 154.19

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Jean wrote:

Q3. Answer : If a such sequence S exists, the number 3 is not inside, and all the terms of S are only equal to 1 and 5 mod 6.

Suppose that the number 3 is inside S .
All the other terms of S are > 3, and, in S , all the terms are ≥ 15 = 3+5+7.
Now, if we work modulo 6, the other terms of T = S (mod 6) are equal to 1 or 5 mod 6.
Locally, in the neighborhood of the number 3, the sequence T looks like this :

T =  d c 3 a b ..

where a = 1 or 5 (mod 6)

If a = 1 (mod 6) then b = 1 (mod 6). (b cannot be 5 since 3+1+5 = 3 (mod 6))

For the same reasons, c = 1 and d = 1 (mod 6) .

Now, d+c+3 = 5 ; c+3+a = 5 ; et 3+a+b = 5

So we have T =  1 1 3 1 1
and T  =    5 5 5

and, with the next step , it will appear a multiple of 3 in T , so T  cannot be a good sequence

We dont need to study the case a = 5. It suffices to replace a,b,c,d with their complements to 6.
--------

When all the primes of S are 1 or 5 (mod 6) i have not found such obstructions.
For example, mod 6, the sequence T = 1, 1, 5, 5 => T ' = 1, 5, 5, 1.=> T ''= 5, 5, 1, 1 , and the chain potentially never end.

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Emmanuel wrote on Feb 3, 2015:

Maybe it is worthwhile to mention that it is very easy to construct long very good necklaces.  I used a kind of greedy algorithm : start with a small necklace  S and expand it to a good necklace with the least prime not in S and so on.
For instance, this allowed me to find a very good  S  with 1000 members (in less than 15 minutes) :

{7,17,19,5,29,37,23,13,47,43,59,127,107,79,293,89,547,173,103,83,167,139,467,113,73,233,157,257,199,227,307
,419,283,479,677,883,149,577,797,349,857,373,617,239,463,269,397,593,409,887,733,719,727,863,359,1777
,563,709,1277,953,499,947,653,1009,1697,313,1613,1283,433,2063,967,773,839,1297,1163,443,787,1193,
523,1487,1303,929,997,2213,1213,1583,1093,1787,853,1409,2417,3823,1907,607,1499,1823,439,4457,1987,
1259,1427,2113,1847,829,2243,859,2087,2357,1459,1877,1423,2729,3307,2333,659,2287,2903,1429,3167,
1447,1709,1753,1559,4177,2687,1129,2633,1069,5717,1993,2129,2347,2753,569,2767,3677,1723,2273,1063,
2777,2003,2089,3467,2393,2833,3557,1543,1979,1597,4217,1399,3593,2137,2879,2683,3389,2647,3797,
2269,3347,3517,1889,2837,4567,1973,2309,5407,4397,1279,4493,2179,6947,2423,1699,5507,3163,2039,
3907,4793,619,7127,3533,1669,6197,2663,3709,5147,5693,3037,3923,2689,7187,2707,2609,3547,4133,
2593,5393,3607,1949,4273,2939,5477,3853,2819,6067,5417,2399,3673,3779,5557,3323,4219,6287,4027,
3299,4003,4049,5827,5483,3559,9497,4933,4943,4153,5273,5077,4349,5503,4229,5237,3433,6173,6967,
2999,4483,5669,4987,6737,3769,5843,4789,7457,5903,4129,8837,4423,4259,5987,5323,3989,6247,7433,
1759,9137,5623,6089,5857,7517,3229,4523,5419,9767,5413,5879,6547,7793,2029,9467,6427,4649,6833,
4999,8087,3943,7883,6977,3583,7559,4597,7817,3919,9413,3319,10607,6397,4679,6373,4919,7927,7283,
1039,7307,7717,5309,7537,9437,4639,8273,4969,10037,7297,5279,6263,6709,9257,6803,4339,12227,
4373,2659,12917,6553,5783,6043,11633,8123,6637,6329,7393,7877,5209,7703,6299,8707,7103,3613,
8423,6883,8093,6337,7829,8513,5479,10247,8147,6277,7649,6343,6029,8807,8167,7019,7853,6529,
12377,7213,8219,8737,13757,4759,10433,4909,10067,8293,10079,9907,13127,6449,9127,7523,6959,
9067,10313,8539,11597,8443,8069,8287,9833,7699,15077,8363,7369,10667,9473,7549,11987,9397,
6779,8573,6679,11447,9157,10169,10463,8599,13877,7583,7993,12983,8623,12647,8839,9203,8209,
13907,9103,7079,15877,10223,7309,16787,8263,10709,11287,11777,5659,11903,10639,14387,11783,
9109,14057,9133,8783,10477,7229,9743,8419,13697,10687,10289,11503,12119,11467,11093,9883,
14153,12277,11807,8329,11657,10333,10589,12697,13313,9719,13267,14423,8779,17327,9697,11909,
8353,10859,13217,8713,10139,15307,13163,8669,12757,12203,9787,10979,10627,13463,8689,14867,
11863,11423,10243,14717,10837,12899,10867,14033,9859,15767,11003,12553,15137,9619,13883,8269,
20747,12577,12503,10453,13397,11069,12433,11789,11677,12413,9949,19727,11593,14009,13807,
15107,11149,12473,7219,17957,10663,11369,12517,12323,8863,15377,11329,14537,11173,14723,
9043,16103,12487,12239,12253,7499,17737,14747,12893,13417,13859,14797,16427,11527,10949,
16963,13499,13567,15443,9439,19937,11443,15263,14173,13553,12073,17093,15083,14713,15773,
13147,12479,13513,14759,17317,13103,10429,22067,14923,17573,11047,15647,13009,17207,
14303,10069,20177,12613,11399,15737,15913,15473,14437,14633,12689,14197,16067,11719,
15527,13687,14699,16453,17159,17377,18233,12799,20897,14827,15149,17167,19013,12829,
16607,14323,16529,15427,17657,12589,16193,11059,23537,14407,20123,10729,18287,18593,
9319,22247,14533,18329,17467,16073,15583,22787,12739,17393,13159,24527,15193,15053,16063,
19157,12409,19997,15217,17099,14083,19949,17827,19913,14519,16987,21377,15559,20333,
17989,25097,17053,16229,16087,19433,12049,23297,17183,15643,20873,16763,16633,21317,
12959,20047,22613,14843,18493,19553,18427,16649,17783,11839,27527,19213,20879,18457,
20693,14149,28547,19183,19559,18517,21323,14479,21587,21727,15749,17443,23057,16033,
18719,21817,21467,13879,20483,15349,25577,16903,21839,21937,22853,14629,22727,19423,
19403,18097,20939,19207,26927,15359,21577,22133,16879,26687,17293,18119,21187,21383,
13669,23957,19447,22283,17419,28517,17713,18773,19417,23063,15889,29717,19483,21713,
18433,23003,17683,19973,21563,19507,22937,18289,24083,15619,31727,19843,19379,20707,
26153,16339,27407,18637,19139,22003,20369,22027,24113,17359,22367,23333,12979,28607,
22013,17659,22877,20407,25349,22343,19819,27917,20023,19739,22147,27767,15289,25343,
20749,26777,19477,25919,21493,20849,25087,26987,19867,22109,23143,24509,23227,24953,
19309,22697,23557,25439,22123,22259,25447,26237,23173,25997,18199,27893,20509,32027,
21163,25013,19753,36083,23017,22409,25147,29123,19777,20639,23603,23029,32297,23663,
21589,34337,20323,27827,22717,23039,25633,18749,27337,25367,23599,29207,24223,24419,
24097,25073,21139,30047,25237,25679,26573,16249,36527,26993,17839,32537,23833,30323,
27043,29387,21499,28493,20149,35747,26293,28619,25537,33287,22039,28283,21649,35117,
22543,25229,28387,25643,23743,26297,25747,23609,22783,26669,26227,32303,21169,34487,
22993,24659,25717,26903,18229,40127,23203,24179,23977,32117,22549,27803,19889,27457,
29303,22129,36107,26393,20719,38327,28597,28793,20089,37847,24547,26489,23887,26693,
27259,35507,26203,26879,30187,31223,17029,38867,25033,29363,27067,29399,23473,27749,
26017,34313,19249,31247,28087,26399,28573,22619,41177,28663,29009,32237,27253,30113,
26407,27779,23893,33029,29287,27743,23789,30367,38543,17599,34847,32887,29339,27793,
26759,29167,31253,25219,34607,27103,33179,32917,33617,22279,39317,26083,28409,28057,
33353,25693,30467,26347,30869,31583,27919,31517,29077,30137,24019,32213,31489,35027,
25423,31793,27943,33767,23719,36563,24109,42257,30637,24989,29587,35573,28027,32003,
27823,32507,23539,39227,30757,33827,24709,37337,30817,32159,31333,30269,28687,31643,
30493,34757,28513,29129,34327,36923,28099,42677,34583,30829,43457,26713,25889,31957,
35267,28549,33563,25579,36137,30553,30089,32587,32693,25849,50387,33623,25609,41597,35227}.

Index (wit base 10 logarithm) :8371.67
Search for such a one with a smaller index could be a nice puzzle.

I forgot to say that you cannot start the procedure with any starting set.
Also, the computing time is much less than 15 minutes.  About 75 seconds are enough for most of the cases I tried.

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