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Probable Prime


A probable prime is a number satisfying Fermat's little theorem (or some other primality test) for some nontrivial base. A probable prime which is shown to be composite is called a pseudoprime; otherwise, it is a (true) prime.

As of May 2024, the largest known probable primes are the repunit primes

R_(8177207)=(10^(8177207)-1)/9
(1)
R_(5794777)=(10^(5794777)-1)/9
(2)

which have 8177207 and 5794777 decimal digits, respectively (Lifchitz and Lifchitz).

Additional large known probable primes include the Wagstaff primes (2^(13347311)+1)/3 and (2^(13372531)+1)/3 (both found by R. Propper in Sep. 2013 and which have 4017941 and 4025533 decimal digits, respectively) and the "dual Sierpinski numbers" 2^n+k (Moore 2009) given by 2^(9092392)+40291 and 2^(5146295)+41693 (which have 2737083 and 1549190 decimal digits, respectively) (Lifchitz and Lifchitz).


See also

Gigantic Prime, Large Number, Primality Certificate, Primality Test, Prime Number, Pseudoprime, Titanic Prime, Wagstaff Prime

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References

Lifchitz, H. and Lifchitz, R. "PRP Records: Probable Primes Top 10000." http://www.primenumbers.net/prptop/prptop.php.Moore, P. "Welcome to 'Five or Bust!' " Oct. 8, 2009. http://www.mersenneforum.org/showthread.php?t=10754.

Referenced on Wolfram|Alpha

Probable Prime

Cite this as:

Weisstein, Eric W. "Probable Prime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ProbablePrime.html

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