Database for k-b-b's:

[science_man_88]

Originally Posted by lorgix

Ok, not following. Explanation?

Anyway, here are the primes.

Should be it for b= 1301 - 1400.

the ones on the diagonal only happen when it's multiplied an even amount of times if not it's on the rim, this is a table of modulo 6 * modulo 6 = modulo 6

the ones highlighted are the only ones b^b can hit so k values are determined but what multiples of these fit things like 6n for 6n+1 on the k*b^b+1 side.

[science_man_88]

Originally Posted by science_man_88

the ones on the diagonal only happen when it's multiplied an even amount of times if not it's on the rim, this is a table of modulo 6 * modulo 6 = modulo 6

the ones highlighted are the only ones b^b can hit so k values are determined but what multiples of these fit things like 6n for 6n+1 on the k*b^b+1 side.

by*

[lorgix]

I've recently started factoring numbers of the form k*b^b+1.

All results are in factordb.

Numbers k*b^b+1 with b and k =<60 , that aren't completely factored:
(sorted by number of digits in composite, then b)
(These range from c95-c108)

6*56^56+1
29*56^56+1
44*58^58+1
54*58^58+1
57*59^59+1
8*60^60+1
48*59^59+1
2*58^58+1
34*58^58+1
27*59^59+1
11*59^59+1
40*60^60+1
44*59^59+1
43*60^60+1
47*60^60+1
54*60^60+1
17*57^57+1
54*57^57+1
13*58^58+1
15*58^58+1
3*59^59+1
38*59^59+1
10*57^57+1
46*60^60+1
14*58^58+1
56*59^59+1
23*58^58+1
23*59^59+1
50*60^60+1
55*60^60+1

Numbers k*b^b+1 with b and k =<75 , that have no known factors:
(sorted by b, then k)

10*57^57+1
7*62^62+1
12*62^62+1
3*66^66+1
28*66^66+1
21*68^68+1
24*69^69+1
49*70^70+1
68*72^72+1
46*73^73+1
18*74^74+1
4*75^75+1
30*75^75+1

[lorgix]

This should be it for k= 2-12000, b= 1401-1500.


I've also continued factoring k*b^b+1, for small k & b.

All results are in factordb.

Particularly interesting is that 10*57^57+1 still has no known factor.

[lorgix]

I've continued factoring numbers k*b^b+1.

I've given potential semi-primes a little extra attention.

I seem to be the only one interested, but just in case I'm off by one or more;

Here's the most interesting find yet:

7*62^62+1 = p54*p59

p54 = 342564991367278926452734188075388370069540741869652959

p59 = 27456160883590352150849440385563209334798961258762792120351

Poly used:

R0: -4072110212622119757654
R1:  456888846229
A0:  10870242064247069137720966187
A1:  1759025599381186974019929
A2: -14527882120011543621
A3: -254910630810921
A4:  1014358042
A5:  8400
skew 113672.30, size 1.376e-010, alpha -7.724, combined = 9.094e-010 rroots = 5

Also noteworthy:

10*57^57+1 = p48*p54
101*54^54+1 = p46*p51

All my results are in factordb.

[3.14159]

Will try and extend this to b = 1700.

Using k = 1 ~ 15000.

1501: 8160, 13086
1502: 4719, 8077, 14358
1503: 1090

[lorgix]

I checked the entire area 2≤k≤30030, 3≤b≤2310 plus and minus one.
Some of the very smallest ones are missing due to the way MultiSieve works.

I have some factors, should anyone want to verify.

Full disclosure: This was done on a 2500K overclocked to 4.2 GHz. It has been perfectly stable for months on end. pfgw didn't report any rounding errors, but there may well be errors of some form or another.

The file contains about 164 thousand primes.