Hello!
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Prime numbers I found so far (all proven by N+1, N-1, or Primo):
731969*3615-1
731969*3617-1
731969*3699-1
731969*2979+1
26350885*365+1
26350885*366+1
26350885*368+1
26350885*3627+1
26350885*3648+1
26350885*3665+1
26350885*36264+1
26350885*36551+1
26350885*362009+1
26350885*362224+1
26350885*364776+1
26350885*365067+1
17944*3618-1 - Sophie Germain - 35888*3618-1
17944*36340-1
17944*361716-1
17944*362666-1
17944*362838-1
4133660484201617547*3618-1
4133660484201617547*3654-1
4133660484201617547*3696-1
4133660484201617547*364209-1
4133660484201617547*3677+1
4133660484201617547*36231+1
4133660484201617547*36882+1
4133660484201617547*361589+1
4133660484201617547*362501+1
833033*365+1
833033*3612+1
833033*36324+1
833033*36647+1
833033*361377+1
833033*364085+1
833033*365204+1
833033*367836+1
833033*3614697+1
50!/(26!*25!)+1
138!/(70!*69!)+1
172!/(87!*86!)+1
180!/(91!*90!)+1
190!/(96!*95!)+1
208!/(105!*104!)+1
290!/(146!*145!)+1
406!/(204!*203!)+1
430!/(216!*215!)+1
454!/(228!*227!)+1
582!/(292!*291!)+1
618!/(310!*309!)+1
644!/(323!*322!)+1
674!/(338!*337!)+1
814!/(408!*407!)+1
960!/(481!*480!)+1
1584!/(793!*792!)+1
1646!/(824!*823!)+1
1838!/(920!*919!)+1
2002!/(1002!*1001!)+1
2318!/(1160!*1159!)+1
4452!/(2227!*2226!)+1
5486!/(2744!*2743!)+1
5644!/(2823!*2822!)+1
8500!/(4251!*4250!)+1
16154!/(8078!*8077!)+1
16218!/(8110!*8109!)+1
32048!/(16025!*16024!)+1
RSA-2048*2164-1
RSA-2048*2226-1
RSA-2048*2834-1
RSA-2048*21732-1
RSA-2048*23920-1
RSA-2048*24176-1
RSA-2048*24220-1
RSA-2048*25756-1
RSA-2048*29250-1
RSA-2048*214832-1
643786079*36^16-1
643786079*36^38-1
643786079*36^99-1
643786079*36^304-1
643786079*36^422-1 (666 digits!)
Numbers I like the most (composites have factors after them):
911
9118675309
8675309
4198862272=2^6*11*5964293
4198862272127
709
907
90709
phi=1.6180339887...
pi=3.1415926535...
M911=1823*26129303*201955048939840841121786425435009*1799520676358263836249116773983093776067731788104797976755522863204273973214237799542669775253771301268617374756896653578445267522764696603459866460147715295527356171742430813085374395309929925098792998479072683867643648045147452807
M43112609
M127, MM127, MMM127, etc. (They might all be prime, but it's extremely unlikely)
Base-36 Values of k:
VELOCIRAPTOR (4133660484201617547=34*173*311*948513010129) (only 3 primes for n<1000, still searching n>1000, k value too big for Proth & NewPGen, using PFGW)
HURT (833033)
FOSH (731969) (a sentient alien bird species; + side is divisible by 5)
FOSH1 (26350885) (same as above)
DUG (17944) (an alien that uses it's arms as legs and vice versa, searched up to n=10000)
Other Base-36 k values I may use in the future:
HUNT (832889) (All composite; + side is divisible by 5, - side is divisible by 7; cofactor could possibly be prime, though)
HUTT (833105=5*7*13*1831)
SKYWALKER (80634324010131=3^2*239*37486900981; - side is divisible by 5)
LUKE (1019390=2*5*101939; - side is divisible by 7)
ANAKIN (643786079=1493*431203) (+ side is divisible by 35)
DARTH (22337621=3251*6871; - side is divisible by 5)
VADER (52552035=3^2*5*1167823)
Some notable Base-94 numbers:
www.mersenne.org (23173464146192129623556375950544=2^4*43*61*2551*11981*2888628773*6254281541)
12345ABCabc`~^&* (403838401633020116561588357495=5*2521*29221*2597002043*422179750373)
V?~17OPfwq3an (15217655020295956113030421=78914411*192837465647382911)
1tRaaRt1 (102991592304415=5*19*43*25212140099)
https://www.google.com/webhp?hl=en (5657439476159105326705063160658143662130016410521642446186695866793; The only prime URL I've found so far.)
QuiGon-Jinn (1432756288327546284743), middle of sexy prime triplet (with QuiGon-Jint (1432756288327546284749) and QuiGon-Jinh (1432756288327546284737))
...and many others. :-)