Just for fun, I looked for small prime factors of the numbers k*2^5523860 + 1, k = 1 to 12. For all but three of them, the smallest factor can be found mentally. For two of the remaining three, the smallest factor is still quite small.
k = 1 p = 17
k = 2 p = 3
k = 3 p = 14270779
k = 4 p = 5
k = 5 p = 3
k = 7 p = 2625617
k = 8 p = 3
k = 9 p = 5
k = 10 p = 11
k = 11 p = 3
k = 12 p = 7
For the remaining value, 6*2^5523860 + 1, I didn't look far enough to find any factors, but I didn't look all that far. Has (as I suspect) someone already found a factor by trial division, or otherwise shown it to be composite?
