Problems & Puzzles: Puzzles

Puzzle 214. Trotter's Curio

Terry Trotter shows the following property of the prime 11:

11: Begin with 11, and continually [i.e. recursively] add the first five powers of 2, but in reverse order (32, 16, , 2). All sums are primes (43, 59, 67, 71, and 73).
 

We may generalize this property and to ask for the least prime p for a given n>0 value, such all the following numbers are primes:

p
p+2n
p+2n+2n-1
p+2n+2n-1+2n-2
...
p+2n+2n-1+2n-2+...22 +2

Here are the results of my own little search for these pairs (n, p):

n p
1 3
2 7
3 5
4 13
5 11 (Trotter)
6 337
7 1889
8 25793653
9 ?
10 ?
11 ?
12 ?

Question: Can you complete the above Table of results?


Solution:

Luke Pebody (N=9 &10) and J. K. Andersen (N=9 to 14) sent contributions to this puzzle.

Here is the Andersen's email

Minimal values of p

n= 9: 13,573,476,641

n=10: 232,317,865,657

n=11: 36,756,785,514,929

n=12: 36,756,785,510,833

n=13: 439,787,787,117,311

n=14: 191,128,877,173,556,587

The n=12 solution is the n=11 solution extended with a smaller prime. All solutions were found with a modified version of the C program written for puzzle 206. n=14 took a day on a 1333 MHz Athlon.

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