R383 is complete to n=25K; 2 primes found for n=5K-25K; 9 k's remaining; base released.

Since this was from my original k=2 search effort and despite only eliminating 2 out of 11 k's after n=5K, I was lucky enough to find:
2*383^20956-1 is prime!

This leaves Riesel bases 170 and 303 as the only 2 bases < 500 where k=2 does not yet have a prime on the Riesel side. R170 is at n=50K and shown on the pages and R303 is at n=10K and not shown. As a base, R303 is a tough one due to its high conjecture of k=85368, although I may search just k=2 for it up to n=25K in the near future.

The Sierp side still has 7 bases < 500 remaining where k=2 does not yet have a prime. 6 out of 7 have been searched to n>=25K with base 383 as the only one not yet there at n=10K. With a conjecture of k=1022, it doesn't appear bad but a prelimiary search to n=2500 showed 64 k's remaining so I haven't been willing to take it on just yet.

One more note of interest on the k=2 search effort: Base 383 had been the lowest base where k=2 remained on both sides. Now the lowest is base 578. k=2 on both sides has been searched to n=10K. Despite low conjectures on both sides, the base is remarkably low weight and hence there are 23 and 10 k's remaining respectfully at n=2500.


Gary

Taking Sierpinski base 484 with conjectured k = 96.

Sierpinski base 484
 

Primes Found:

[code]
3*484^1+1
4*484^3+1
7*484^1+1
9*484^1+1
10*484^16+1
12*484^2+1
15*484^57+1
16*484^10+1
18*484^1+1
19*484^5+1
21*484^1060+1
24*484^1+1
25*484^1+1
28*484^1+1
30*484^41+1
31*484^8+1
33*484^1+1
36*484^204+1
37*484^1+1
39*484^33+1
40*484^3+1
42*484^6+1
43*484^2+1
46*484^2+1
49*484^3+1
51*484^4+1
52*484^1+1
57*484^8+1
58*484^6+1
60*484^8+1
61*484^10+1
63*484^1+1
64*484^1+1
66*484^24+1
67*484^1+1
70*484^10+1
72*484^1+1
73*484^4+1
75*484^4+1
78*484^864+1
79*484^1+1
81*484^16+1
82*484^2+1
84*484^103+1
85*484^1+1
87*484^12+1
88*484^27+1
93*484^1+1
94*484^1+1
[/code]

k=1 is a GFN. I have not tested it.

k=54 remains and has been tested to n=25000.

The other k have trivial factors.

I will continue to work on this conjecture. Maybe I'll get lucky and knock off another conjecture that has a single k remaining.

Reserving R328 to 100K

R321 complete to 100K, releasing base.

[quote=Batalov;207606]Reserving R328 to 100K[/quote]
Now, 41*328^31734-1 showed up prime and promoted R328 to the one-k club. Going on...

Reserving Riesel Bases 469 and 499 as new to n=25K.

R294 is proven (with a pencil)
 

b=294=6*7[sup]2[/sup]

k=6: for even n, divisible by 5; for odd n=2m-1,
6*294[sup]2m-1[/sup]-1 = 6*(6*7^2)[sup]2m-1[/sup]-1 = 6[sup]2m[/sup] * 7[sup]2n[/sup] - 1[sup]2[/sup] = Difference of squares.

k=96=6*4[sup]2[/sup]: ditto.

qed :smile:

R288
 

[I]b[/I]=288 = [B]2[/B][sup][B]5[/B][/sup]*3[sup]2[/sup]
[I]k[/I]=18 = [B]2[/B]*3[sup]2[/sup]
[I]k[/I]=392 = [B]2[/B][sup][B]3[/B][/sup]*7[sup]2[/sup]
For both [I]k[/I] and even [I]n[/I], trivial factors, for odd [I]n[/I], we have differences of squares.

Reserving to 50K (two k remaining).

R444, [I]k[/I]=111 eliminates (algebraic with [I]n[/I] odd, trivial with even)...

There's probably quite a few more of these.

One last proof before going to sleep:

R414 is proven (k=46 = b/3[SUP]2[/SUP]); the rest is ditto.