[QUOTE=sweety439;531552]A (probable) prime was found:

(13998*40^12381+29)/13

Written in base 40, this number is Qa{U[SUB]12380[/SUB]}X

This number is likely the second-largest "base 40 minimal prime"[/QUOTE]

Another probable prime is (13998*40^13474+29)/13, but this is not minimal prime in base 40

[QUOTE=sweety439;531730]Base 36:

O{L}Z (4428*36^n+67)/5: tested to n=50K, no (probable) prime found
{P}SZ (6480*36^n+821)/7: currently at n=41566, no (probable) prime found

Base 40:

S{Q}d (86*40^n+37)/3: currently at n=59777, no (probable) prime found[/QUOTE]

Base 36 {P}SZ (6480*36^n+821)/7 tested to n=50K, no (probable) prime found.

Result file attached.

Base 40:

S{Q}d (86*40^n+37)/3: currently at n=87437, no (probable) prime found

Unsolved families:

Base 17:

F1{9}: (4105*17^n-9)/16

Base 19:

EE1{6}: (15964*19^n-1)/3

Base 21:

G{0}FK: 7056*21^n+335

Base 25:

EF{O}: 366*25^n-1
O{L}8: (4975*25^n-111)/8
CM{1}: (7729*25^n-1)/24
E{1}E: (8425*25^n+311)/24
EE{1}: (8737*25^n-1)/24
6M{F}9: (34525*25^n-53)/8
F{1}F1: (225625*25^n+8399)/24

Base 26:

{A}6F: (1352*26^n-497)/5
{I}GL: (12168*26^n-1243)/25

Base 31:

E8{U}P: 13733*31^n-6
{F}RA: (961*31^n+733)/2
{F}G: (31*31^n+1)/2
{F}KO: (961*31^n+327)/2
IE{L}: (5727*31^n-7)/10
{L}G: (217*31^n-57)/10
{L}CE: (6727*31^n-2867)/10
M{P}: (137*31^n-5)/6
{P}I: (155*31^n-47)/6
{R}1: (279*31^n-269)/10
{R}8: (279*31^n-199)/10
{U}P8K: 29791*31^n-5498

Base 31:

ILE{L}: (179637*31^n-7)/10 [need not to be searched if a smaller prime for the "IE{L}: (5727*31^n-7)/10" family were found]
L{F}G: (1333*31^n+1)/2 [need not to be searched if a smaller prime for the "{F}G: (31*31^n+1)/2" family were found]
L0{F}G: (40393*31^n+1)/2 [need not to be searched if a smaller prime for either the "{F}G: (31*31^n+1)/2" family or the "L{F}G: (1333*31^n+1)/2" family were found]
{L}9G: (6727*31^n-3777)/10 [need not to be searched if a smaller prime for the "{L}G: (217*31^n-57)/10" family were found]
{L}9IG: (208537*31^n-116307)/10 [need not to be searched if a smaller prime for either the "{L}G: (217*31^n-57)/10" family or the "{L}9G: (6727*31^n-3777)/10" family were found]
{L}SO: (6727*31^n+2193)/10
{L}IS: (6727*31^n-867)/10
MI{O}L: (108624*31^n-19)/5
P{F}G: (1581*31^n+1)/2 [need not to be searched if a smaller prime for the "{F}G: (31*31^n+1)/2" family were found]
PEO{0}Q: 758973*31^n+26
{R}1R: (8649*31^n-8069)/10 [need not to be searched if a smaller prime for the "{R}1: (279*31^n-269)/10" family were found]
SP{0}K: 27683*31^n+20

Base 35:

6W{P}4: (288855*35^n-739)/34 [need not to be searched if a smaller prime for the "W{P}4: (38955*35^n-739)/34" family were found]
F8{0}F9: 652925*35^n+534
{Y}PO: 1225*35^n-326
FQ{F}I: (656215*35^n+87)/34
PX{0}ER: 1112300*35^n+517
Q{P}4: (31815*35^n-739)/34
RF{0}CPI: 41160000*35^n+15593

Base 36:

O{L}Z: (30996*36^n+469)/35
{P}SZ: (6480*36^n+821)/7

Base 40:

S{Q}d: (3440*40^n+37)/3

Base 25:

F{O}KO: 9375*25^n+524
FO{K}O: (56375*25^n+19)/6
LO{L}8: (109975*25^n-111)/8 [need not to be searched if a smaller prime for the "O{L}8: (4975*25^n-111)/8" family were found]
M{1}F1: (330625*25^n+8399)/24
M1{0}8: 13775*25^n+8

Base 28:

O{A}F: (18424*28^n+125)/27

Base 35:

LAA{E}6: (15520820*35^n-143)/17
{L}E6: (25725*35^n-8861)/34
P0{P}G: (1042125*35^n-331)/34
{Q}PEM: (557375*35^n-28046)/17
RU{A}C: (580300*35^n+29)/17
W{P}4: (38955*35^n-739)/34
{X}MLX: (1414875*35^n-472463)/34
X{M}Y: (20020*35^n+193)/17

Base 27:

8{0}9A: 5832*27^n+253
999{G}: (88577*27^n-8)/13
C{L}E: (8991*27^n-203)/26
E{I}F8: (139239*27^n-1192)/13
{F}9FM: (295245*27^n-113557)/26

Base 48:

A{0}SP: 23040*48^n+1369
C{e}Z: (28992*48^n-275)/47
{K}IP: (46080*48^n-4297)/47
a{0}1: 1728*48^n+1
eL{0}Z: 93168*48^n+35
jc{e}Z: (4960608*48^n-275)/47

The "minimal prime problem" is solved only in bases 2~16, 18, 20, 22~24, 30, 42, and maybe 60

[CODE]
b, length of largest minimal prime base b, number of minimal primes base b
2, 2, 2
3, 3, 3
4, 2, 3
5, 5, 8
6, 5, 7
7, 5, 9
8, 9, 15
9, 4, 12
10, 8, 26
11, 45, 152
12, 8, 17
13, 32021, 228
14, 86, 240
15, 107, 100
16, 3545, 483
18, 33, 50
20, 449, 651
22, 764, 1242
23, 800874, 6021
24, 100, 306
30, 1024, 220
42, 487, 4551
60, 1938, ?
[/CODE]

Some minimal (probable) primes with bases 28<=b<=50 not shown in [URL="https://github.com/RaymondDevillers/primes"]https://github.com/RaymondDevillers/primes[/URL]: (and hence some unsolved families can be removed)

Base 37: (families FY{a} and R8{a} can be removed)

590*37^22021-1 (= FY{a_22021})
1008*37^20895-1 (= R8{a_20895})

Base 40: (family Qa{U}X can be removed)

(13998*40^12381+29)/13 (= Qa{U_12380}X)

Base 45: (families O{0}1 and AO{0}1 can be removed, and hence families O{0}1F1, O{0}ZZ1, unless they have small (probable) primes)

24*45^18522+1 (= O{0_18521}1)
474*45^44791+1 (= AO{0_44790}1) [this prime is not minimal prime]

Base 49: (families 11c{0}1, Fd{0}1, SL{m} and Yd{m} can be removed, and hence families S6L{m}, YUUd{m}, YUd{m}, unless they have small (probable) primes)

2488*49^29737+1 (= 11c{0_29736}1)
774*49^18341+1 (= Fd{0_18340}1)
1394*49^52698-1 (= SL{m_52698})
1706*49^16337-1 (= Yd{m_16337})

Although the test limit of all families in [URL="https://github.com/RaymondDevillers/primes"]https://github.com/RaymondDevillers/primes[/URL] are all 10K, but some families are in fact already tested to much higher....

Base 25:

EF{O}, 366*25^n-1: 260K, see [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm[/URL]

Base 31:

F{G}, (1*31^n+1)/2: 2^19-2, see [URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL]

Base 32:

4{0}1, 4*32^n+1: (2^33-7)/5, see [URL="http://www.prothsearch.com/fermat.html"]http://www.prothsearch.com/fermat.html[/URL]
G{0}1, 16*32^n+1: (2^34-9)/5, see [URL="http://www.prothsearch.com/fermat.html"]http://www.prothsearch.com/fermat.html[/URL]
UG{0}1, 976*32^n+1: 560K, see [URL="http://www.prothsearch.com/riesel1.html"]http://www.prothsearch.com/riesel1.html[/URL]

Base 38:

1{0}1 1*38^n+1: 2^24-2, see [URL="http://www.primegrid.com/stats_genefer.php"]http://www.primegrid.com/stats_genefer.php[/URL]

Base 45:

9W1{0}1 19666*45^n+1: 100K, see [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm[/URL]

Base 46:

d4{0}1, 1798*46^n+1: 500K, see [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm[/URL]

Base 48:

a{0}1, 36*48^n+1: 500K, see [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm[/URL]

Base 50:

1{0}1 1*50^n+1: 2^24-2, see [URL="http://www.primegrid.com/stats_genefer.php"]http://www.primegrid.com/stats_genefer.php[/URL]