--- In primeform@yahoogroups.com, "David Broadhurst"
<d.broadhurst@...> wrote:

> a[n] is smallest number such that
> (a+1)^(2^n)+1 and
> (a-1)^(2^n)+1 are both prime

I was lucky with a[11]. Up to n=12 the sequence is

3, 3, 3, 75, 113, 2163, 63739, 13221, 54809, 3656571,
6992033, 125441

assuming that Yves and I missed no smaller prime pairs.

(3+1)^(2^1)+1 is prime! (0.0023s+0.0078s)
(3-1)^(2^1)+1 is prime! (0.0023s+0.0094s)
(3+1)^(2^2)+1 is prime! (0.0027s+0.0089s)
(3-1)^(2^2)+1 is prime! (0.0024s+0.0078s)
(3+1)^(2^3)+1 is prime! (0.0027s+0.0091s)
(3-1)^(2^3)+1 is prime! (0.0024s+0.0070s)
(75+1)^(2^4)+1 is prime! (0.0241s+0.0103s)
(75-1)^(2^4)+1 is prime! (0.0071s+0.0077s)
(113+1)^(2^5)+1 is prime! (0.0127s+0.0096s)
(113-1)^(2^5)+1 is prime! (0.0072s+0.0078s)
(2163+1)^(2^6)+1 is prime! (0.0500s+0.0110s)
(2163-1)^(2^6)+1 is prime! (0.0162s+0.0073s)
(63739+1)^(2^7)+1 is prime! (0.0770s+0.0106s)
(63739-1)^(2^7)+1 is prime! (0.0767s+0.0074s)
(13221+1)^(2^8)+1 is prime! (0.2505s+0.0111s)
(13221-1)^(2^8)+1 is prime! (0.2479s+0.0096s)
(54809+1)^(2^9)+1 is prime! (1.2217s+0.0101s)
(54809-1)^(2^9)+1 is prime! (0.8976s+0.0080s)
(3656571+1)^(2^10)+1 is prime! (5.3727s+0.0108s)
(3656571-1)^(2^10)+1 is prime! (5.3709s+0.0076s)
(6992033+1)^(2^11)+1 is prime! (25.8867s+0.0139s)
(6992033-1)^(2^11)+1 is prime! (25.8388s+0.0064s)
(125441+1)^(2^12)+1 is prime! (50.1531s+0.0154s)
(125441-1)^(2^12)+1 is prime! (49.8836s+0.0073s)

David