(1035-1)/9 + 2*1034 + 2*1017 + 2
Using Pari/gp (< 1s)
     gp > isprime((10^35-1)/9+2*10^34+2*10^17+2)
     %1 = 1


(1089-1)/9 + 2*1088 + 2*1044 + 2
Using Pari/gp (< 1s)
     gp > isprime((10^89-1)/9+2*10^88+2*10^44+2)
     %1 = 1


(10175-1)/9 + 2*10174 + 2*1087 + 2
Using Pari/gp (< 1s)
     gp > isprime((10^175-1)/9+2*10^174+2*10^87+2)
     %1 = 1


(102309-1)/9 + 2*102308 + 2*101154 + 2
Using Wolframalfa
     Factor (10^2309-1)/9+2*10^2308+2*10^1154+2
     3111...1113111...1113 (prime number)

Using Pari/gp (50min, 29,671 ms)
     gp > default(parisizemax,"1G")
     gp > isprime((10^2309-1)/9+2*10^2308+2*10^1154+2)
     %1 = 1

Proven by certificate by Masaki UKAI (August 22, 2022)
http://factordb.com/index.php?id=1100000002732284810
 

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