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# 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.

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## Mathematics Subject Classification

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