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MCL: Model Checking Language; Alternation-Free Modal μ-calculus extended with user-friendly regular expressions and value-passing constructs; subsumes CTL and LTL. mCRL2 mu-calculus: Kozen's propositional modal μ-calculus (excluding atomic propositions), extended with: data-depended processes, quantification over data types, multi-actions ...
Randomly pick a number a. Check equality (corresponding to the chosen test) involving a and the given number n. If the equality fails to hold true, then n is a composite number and a is a witness for the compositeness, and the test stops. Get back to the step one until the required accuracy is reached.
Input #1: b, the number of bits of the result Input #2: k, the number of rounds of testing to perform Output: a strong probable prime n while True: pick a random odd integer n in the range [2 b −1 , 2 b −1] if the Miller–Rabin test with inputs n and k returns “ probably prime ” then return n
A strong pseudoprime is a composite number that passes the Miller–Rabin primality test. All prime numbers pass this test, but a small fraction of composites also pass, making them " pseudoprimes ". Unlike the Fermat pseudoprimes , for which there exist numbers that are pseudoprimes to all coprime bases (the Carmichael numbers ), there are no ...
Using fast algorithms for modular exponentiation and multiprecision multiplication, the running time of this algorithm is O(k log 2 n log log n) = Õ(k log 2 n), where k is the number of times we test a random a, and n is the value we want to test for primality; see Miller–Rabin primality test for details.
There is also overlap among strong pseudoprimes to different bases. For example, 1373653 is the smallest strong pseudoprime to bases 2 through 4, and 3215031751 is the smallest strong pseudoprime to bases 2 through 10. Arnault [11] gives a 397-digit Carmichael number N that is a strong pseudoprime to all prime bases less than 307.
The safety added by bounds checking necessarily costs CPU time if the checking is performed in software; however, if the checks could be performed by hardware, then the safety can be provided "for free" with no runtime cost. An early system with hardware bounds checking was the ICL 2900 Series mainframe announced in 1974. [3]
The following program in Python determines whether an integer number is a Munchausen Number / Perfect Digit to Digit Invariant or not, following the convention =. num = int ( input ( "Enter number:" )) temp = num s = 0.0 while num > 0 : digit = num % 10 num //= 10 s += pow ( digit , digit ) if s == temp : print ( "Munchausen Number" ) else ...