Problems & Puzzles: Puzzles

Puzzle 77.- Christmas Prime-Pine.

Regarding the pine shown below, each sphere (X) is in contact with two (Y & Z) in the line below, such that:

a) X = (Y + Z)/2

b) X, Y, & Z are prime numbers.

c)All numbers in the pine must be distinct prime numbers (except in the question 4)

Questions

1. Find the least assignation of primes for a Prime-Pine of R rows, for R>8 to…

2. In particular find the least solution for R = 36 and win the Prime Devil Prize.

3. A one color - prime pine, R=5,  with all the ending digits equal to "1" is the one with the last row being 11, 71,1571,1151 & 251. Find a larger one color - prime pine ending in the same D-ending digit (D=1, 3, 7 or 9).

4. Can you develop a method for making the least assignation to a pine of R rows, using distinct natural numbers?

(Merry Christmas and Happy New Year, folks…!!!)

Jacques Tramu wrote (on March 09, so 10 years is nothing!):

I Checked your solution is optimal for R = 8
Here is a solution for R = 9

27479
33809 21149
40949 26669 15629
48299 33599 19739 11519
54869 41729 25469 14009 9029
58889 50849 32609 18329 9689 8369
57119 60659 41039 24179 12479 6899 9839
42989 71249 50069 32009 16349 8609 5189 14489
227 85751 56747 43391 20627 12071 5147 5231 23747

R=9 Total:1308783

***

Jan van Delden wrote (Nov 2012):

Couldn’t find the 10 solution yet, maybe my algorithm(s) can be made faster. Have to think about that a bit more.
The Q3 bit could probably be extended to R=8/9, giving a bit a more time.

My contribution:

Q3:

Solutions with R=7, bottom rows (not necessarily the least assignation):

Ending digit 1: 9431,15791,5471,5591,8231,1151,4271

Ending digit 3: 10343,12143,4583,1823,743,4943,2663

Ending digit 7: 11677,12517,9157,8317,397,757,10357

Ending digit 9: 59,8039,1259,119,2939,5639,7019

Q4:

The “least assignation”-part makes this question tough.

A pattern for a solution with the numbers a[i] at the bottom row sorted:

a[i]=2*i*(i+1), i in [0,R-1].
The already given solutions suggest that this will not generate a least assignation.

Any assignation in integers, a[i,j] with i on [0,R-1] and j on [0,i], for the whole pine,

will generate a prime solution if:

- the a[i,j] are even (if not multiply everything with 2)

- are admissible, i.e. don’t form a complete residue set for alle primes p<R*(R+1)/2.

For instance if the bottom row should be sorted, one could prove that 3 can not be part of the bottom of the tree and hence the whole tree (except when R=2) and 12 is a divisor of the differences of the primes at the bottom.

Example bottom row (R=5):

solution 1 mod 3: 751,487,439,127,31

solution 2 mod 3: 821,701,653,389,5

***

Jan van Delden wrote on May 07, 2015:

Q1

The solution with R=9 having smallest sum 673791, with p<82781.

8831
8243 9419
8969 7517 11321
11177 6761 8273 14369
14771 7583 5939 10607 18131
19463 10079 5087 6791 14423 21839
25919 13007 7151 3023 10559 18287 25391
37217 14621 11393 2909 3137 17981 18593 32189
67421 7013 22229 557 5261 1013 34949 2237 62141

I found 19 solutions in this range.

Q3:

I was able to extend the solutions to R=9.
The solutions with smallest maximum prime:

77611
71821 83401
65881 77761 89041
63391 68371 87151 90931
68611 58171 78571 95731 86131
82891 54331 62011 95131 96331 75931
101611 64171 44491 79531 110731 81931 69931
117841 85381 42961 46021 113041 108421 55441 84421
139021 96661 74101 11821 80221 145861 70981 39901 128941
The sum is equal to: 3622635

98873
102653 95093
107843 97463 92723
113453 102233 92693 92753
118673 108233 96233 89153 96353
123833 113513 102953 89513 88793 103913
126143 121523 105503 100403 78623 98963 108863
108803 143483 99563 111443 89363 67883 130043 87683
75773 141833 145133 53993 168893 9833 125933 134153 41213
The sum is equal to: 4601985

38377
37357 39397
41227 33487 45307
48397 34057 32917 57697
54907 41887 26227 39607 75787
56737 53077 30697 21757 57457 94117
54277 59197 46957 14437 29077 85837 102397
60397 48157 70237 23677 5197 52957 118717 86077
118927 1867 94447 46027 1327 9067 96847 140587 31567
The sum is equal to: 2358735

89809
92899 86719
97729 88069 85369
101929 93529 82609 88129
100999 102859 84199 81019 95239
91369 110629 95089 73309 88729 101749
75289 107449 113809 76369 70249 107209 96289
64189 86389 128509 99109 53629 86869 127549 65029
76519 51859 120919 136099 62119 45139 128599 126499 3559
The sum is equal to: 4041225

***

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