Just now I was doing an off-the-wall search for MooMoo's "BEEF15BAD" residue with very small numbers (a fixed n search for k=3-1G, n=5, k*2^n+1), when I noticed a very weird pattern:

Code:

168195*2^5+1 = 5382241 is prime! (trial divisions)
168221*2^5+1 = 5383073 is prime! (trial divisions)
168225*2^5+1 = 5383201 is prime! (trial divisions)
168231*2^5+1 = 5383393 is prime! (trial divisions)
168239*2^5+1 = 5383649 is prime! (trial divisions)
168245*2^5+1 = 5383841 is prime! (trial divisions)
168251*2^5+1 = 5384033 is prime! (trial divisions)
168269*2^5+1 = 5384609 is prime! (trial divisions)
168281*2^5+1 = 5384993 is prime! (trial divisions)
168293*2^5+1 = 5385377 is prime! (trial divisions)
168305*2^5+1 = 5385761 is prime! (trial divisions)

This continues such that every number so far in my sieve file is prime. (I stopped the LLR testing at that point since I figured it was a pointless exercise, considering that it wouldn't be able to produce any composites and therefore useful residues.)

I'm sure there's a simple mathematical explanation for what I'm seeing here. You know, though I hate to sound "crankish"...if there is a simple mathematical proof that all k*2^5+1 are prime, then this could lead to a very simple way to find a 100 million digit prime that would qualify for the EFF prize! Heck on spending 3+ years per number searching 100 million digit numbers through GIMPS when you can just find one this way.

(Of course, I'm sure there's something I'm missing that would preclude this, otherwise someone would have won the prize by now.)

Edit: I'm seeing this on k*2^7+1 as well.