4167*2^1836466-1 (552835 digits)

[QUOTE=Trilo;413496]1423*2[SUP]2178363[/SUP]- 1 is prime (655,756 digits)
1423*2[SUP]2179023[/SUP]- 1 is prime (655,955 digits)

Very interesting small gap between the 2 primes with n so large. k=1423 also has a very low nash weight of 309.[/QUOTE]

Just my 2 cents...

69 has this too as it seems - 2 primes close to each other.

So average branching factor might seem good at 1.2 roughly, yet because you find 2 primes at once, branching factor to find next prime is more like factor 1.44

So where finding 2 primes very close to each is great news for now for you - it might not necessarily be good news for the long run if you continue to search the formula.

Do you know what "independent events" means in the context of prime searching? Finding two primes, or no primes, does nothing to alter the frequency of future primes in your search.

Your comments are akin to saying "nice finds, but you've used up your probability!"

With 69 as it seems probability is higher that 2 primes are close to each other.

Regrettably already testing at nearly 4.3M that seems to be true.

As 1.2 * last found prime = 3.14M = 3.8M and there is like a handful of exponents left at 3.99M after which next odds is a few exponents around 4.22M - 4.30M

So no matter how much i do hope your statement to be true - i don't believe it for specific formula's such as 69 at millions of bits range.

You should realize more than anyone else that if the search space becomes huge enough that luck plays less of a role and that a given heuristic performs as it should be.

Please note that the implications of this lemma are far further reaching than just a few primes.

Amongst others the odds for life in the universe and also the reason why our brain can work as it does work.

*Note that for primes the valid question you could ask is whether testing just a couple of hundreds of thousands of exponents gives a big enough search space for randomness to play a smaller role...

In Game Tree search the break even point seems more in the range of tens of millions of nodes rather than couple of hundreds of thousands.

A bunch of primes for low weight Ks:
269742256597*2^1340989-1 (403690 digits)
2444379546449*2^1699964-1 (511753 digits)
2135489665061*2^1929362-1 (580809 digits)

4071*2^1721361-1 (518185 digits)

1246461300659*2^2103424-1 (633206 digits)

4089*2^1803463-1 (542901 digits)

Low weights again - this time submitted with the proper proof-code...

3903177334109*2^1330952-1 (400670 digits)
3882354543517*2^1387081-1 (417566 digits)

4111*2^1754463-1 (528150 digits)