examples of probable primes

To give an example of a probable prime relative to a base: $4^{341233}-3^{341233}$ has passed preliminary primality tests relative to bases 2, 3, 5, 7, 11, 13 and 101. Its square root is approximately $2.3362\cdot {10}^{102721}$, which makes a conclusive primality test by trial division in a reasonable time period impractical.

To give an example of a probable prime by a pattern: this pattern

$$2^2-1=3,2^3-1=7,2^7-1=127$$

$$2^{127}-1=170141183460469231731687303715884105727$$

suggests that $2^{170141183460469231731687303715884105727}-1$ might be a Mersenne prime. But since this is larger than the largest known Mersenne prime $2^{30402457}-1$ (as of 2005), a Lucas-Lehmer test might take longer than the average human lifetime.

On the other hand, $123456789\cdot {10}^{123456789}+123456789$ is not a probable prime, because even though it is much larger than either of the probable primes given above, it is clearly divisible by $3^2$.