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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