Using Pari/gp (< 1s)
     gp > isprime(4*(10^7-1)/9-10^6-10^3-1)
     %1 = 1

Using Pari/gp (< 1s)
     gp > isprime(4*(10^11-1)/9-10^10-10^5-1)
     %1 = 1

Using Pari/gp (< 1s)
     gp > isprime(4*(10^17-1)/9-10^16-10^8-1)
     %1 = 1

Using Pari/gp (< 1s)
     gp > isprime(4*(10^47-1)/9-10^46-10^23-1)
     %1 = 1

Using Pari/gp (< 1s)
     gp > isprime(4*(10^55-1)/9-10^54-10^27-1)
     %1 = 1

Using Pari/gp (time = 2,234 ms)
     gp > isprime(4*(10^367-1)/9-10^366-10^183-1)
     %1 = 1

Using Pari/gp (time = 3,314 ms)
     gp > isprime(4*(10^413-1)/9-10^412-10^206-1)
     %1 = 1

Using Pari/gp (1min, 22,328 ms)
     gp > default(parisizemax,"1G")
     gp > isprime(4*(10^955-1)/9-10^954-10^477-1)
     %1 = 1

Proven by certificate by Masaki UKAI (October 21, 2022)
http://factordb.com/index.php?id=1100000002732285048

C:\pfgw>pfgw64 -q"4*(10^25487-1)/9-10^25486-10^12743-1"
PFGW Version 4.0.4.64BIT.20221214.Win_Dev [GWNUM 30.11]

4*(10^25487-1)/9-10^25486-10^12743-1 is 3-PRP! (3.2744s+0.0004s)

C:\pfgw>pfgw64 -q"4*(10^34489-1)/9-10^34488-10^17244-1"
PFGW Version 4.0.4.64BIT.20221214.Win_Dev [GWNUM 30.11]

4*(10^34489-1)/9-10^34488-10^17244-1 is 3-PRP! (7.0442s+0.0008s)