Problems & Puzzles: Puzzles

Puzzle 815. Another prime pine

Jan van Delden sent the following nice puzzle. It's a new version of the old (1999) Puzzle 77 from my pages:

Letís define our Christmas prime tree. See  Puzzle 77 for a different kind of tree.

The tree has n rows of baubles. All baubles are decorated with different prime numbers.
 
As a non-minimal example (n=6):
 
2141
1493, 2789
257, 2729, 5381
71, 443, 7673, 13337
29, 113, 1187, 29363, 35993
5, 53, 281, 4409, 142067, 69143
 
Largest prime number: 142067
Our peak:                      2141
With sum:                     318957
 
 
We have: 
 
29 = (5+53)/2, 113=(5+53+281)/3, 1187=(5+53+281+4409)/4,
29363=(5+53+281+4409+142067)/5,35993=(5+53+281+4409+142067+69143)/6
 
So the prime numbers on a row are equal to the moving average of the prime numbers on the row below.
This moving average always starts at the prime on the far left side of the tree. In the example the number 5.
 
Try to answer the following questions using as large a tree as you possibly can!
 
Q1: Find a tree where the largest prime number is minimal.
 
Q2: Find a tree where the prime number at the peak is minimal.
 
Q3: Find a tree for which the sum of all primenumbers is minimized.
 
Q4: Is it possible to find trees where the ending digit D of all the primenumbers are equal?

 


Contributopns came from Emmanuel Vantieghem, Vicente Felipe Izquierdo,

***

Emmanuel wrote:

For n = 6, the tree with minimal maximal prime is :
677
653  701
 
587  719  797
 
347  827  983  1031
 
11  683  1787  1451  1223
 
3   19   2027   5099   107   83
 

There is another tree with smaller top :

 

593

467   719
 
191   743   1223
 
23   359   1847   2663
 
5   41   1031   6311   5927
 
3   7   113   4001   27431   4007
 
but it is possible that there exist a tree with a smaller top.

 
I did not try to minimize the sum of the primes (I think that the above solution could be it). Instead, I started the search for n = 7 and found among all the trees with 3 in the left lower corner the one with smallest maximal prime :
 

6569

6269  6869
 
5639  6899  8069
 
4259  7019  9419  6899
 
569  7949  12539  16619  11579
 
107  1031  22709  26309  32939  38219
 
3   211   2879   87743   40709   66089  69899
 

 
Finding the minimal top tree or the minimal sum tree would cost me at least a week of computing.

***

Vicente wrote:

(n=6)

 

Q1:

677

         

653

701

       

587

719

797

     

347

827

983

1031

   

11

683

1787

1451

1223

 

3

19

2027

5099

107

83

 

Q2:

421

         

271

571

       

193

349

1171

     

19

367

661

3637

   

7

31

1063

1543

15541

 

3

11

79

4159

3463

85531

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

                   

Q3: same tree in Q1

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