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Conway and Kochen, The Strong Free Will Theorem, published in Notices of the AMS. Volume 56, Number 2, February 2009. Rehmeyer, Julie (August 15, 2008). "Do Subatomic Particles Have Free Will?". Science News. Introduction to the Free Will Theorem, videos of six lectures given by J. H. Conway, Mar. 2009. Wüthrich, Christian (September 2011).
For example, 1093 2 = 1194649 is a Fermat pseudoprime to base 2, and 11 2 = 121 is a Fermat pseudoprime to base 3. The number of the values of b for n are (For n prime, the number of the values of b must be n − 1, since all b satisfy the Fermat little theorem)
both 3 and 12 are quadratic residues mod q (per law of quadratic reciprocity) neither 3 nor 12 is a primitive root of q; the only safe primes that are also full reptend primes in base 12 are 5 and 7; q divides 3 (q−1)/2 − 1 and 12 (q−1)/2 − 1, same as 3 (q−1)/2 ≡ 1 mod q and 12 (q−1)/2 ≡ 1 mod q (per Euler's criterion)
Carmichael λ function: λ(n) for 1 ≤ n ≤ 1000 (compared to Euler φ function). In number theory, a branch of mathematics, the Carmichael function λ(n) of a positive integer n is the smallest positive integer m such that
The free will theorem says that if we have free will, then particles must have free will. This presumably is counterintuitive. It makes no claim about a world in which we don't have free will (a deterministic world). There's no way to argue for free will on the basis of this theorem - and yet, this is what the section claims, without any ...
They are called the strong law of large numbers and the weak law of large numbers. [16] [1] Stated for the case where X 1, X 2, ... is an infinite sequence of independent and identically distributed (i.i.d.) Lebesgue integrable random variables with expected value E(X 1) = E(X 2) = ... = μ, both versions of the law state that the sample average
Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) holding times that have finite mean. A renewal-reward process additionally has a random sequence of ...
Later in his life he was a chancellor of Oxford. He was the first to discover the mean-speed theorem, later "The Law of Falling Bodies". Unlike Bradwardine's theory, the theorem, also known as "The Merton Rule" is a probable truth. [15] His most noted work was Regulae Solvendi Sophismata (Rules for Solving Sophisms).