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This picture shows clockwise from top left: An Arithmometer, a Comptometer, a Dalton adding machine, a Sundstrand, and an Odhner Arithmometer. A mechanical calculator, or calculating machine, is a mechanical device used to perform the basic operations of arithmetic automatically, or (historically) a simulation such as an analog computer or a ...
We can find quadratic residues or verify them using the above formula. To test if 2 is a quadratic residue modulo 17, we calculate 2 (17 − 1)/2 = 2 8 ≡ 1 (mod 17), so it is a quadratic residue. To test if 3 is a quadratic residue modulo 17, we calculate 3 (17 − 1)/2 = 3 8 ≡ 16 ≡ −1 (mod 17), so it is not a quadratic residue.
Discriminant of an algebraic number field. A fundamental domain of the ring of integers of the field K obtained from Q by adjoining a root of x3 − x2 − 2 x + 1. This fundamental domain sits inside K ⊗ Q R. The discriminant of K is 49 = 7 2. Accordingly, the volume of the fundamental domain is 7 and K is only ramified at 7.
If each book had a mass of 100 grams, all of them would have a total mass of 10 93 kilograms. In comparison, Earth's mass is 5.972 × 10 24 kilograms, the mass of the Milky Way galaxy is estimated at 2.5 × 10 42 kilograms, and the total mass of all the stars in the observable universe is estimated at 2 × 10 52 kg.
Fermat's little theorem. In number theory, Fermat's little theorem states that if p is a prime number, then for any integer a, the number ap − a is an integer multiple of p. In the notation of modular arithmetic, this is expressed as. For example, if a = 2 and p = 7, then 27 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7 .
1 3 = 1 up 1; 2 3 = 8 down 3; 3 3 = 27 down 1; 4 3 = 64 down 3; 5 3 = 125 up 1; 6 3 = 216 up 1; 7 3 = 343 down 3; 8 3 = 512 down 1; 9 3 = 729 down 3; 10 3 = 1000 up 1; There are two steps to extracting the cube root from the cube of a two-digit number. For example, extracting the cube root of 29791. Determine the one's place (units) of the two ...
Euler's theorem. In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if n and a are coprime positive integers, then is congruent to modulo n, where denotes Euler's totient function; that is. In 1736, Leonhard Euler published a proof of Fermat's little theorem [1] (stated by Fermat ...
Super PI finishing a calculation of 1,048,576, or 2 20 digits of pi. Super PI is a computer program that calculates pi to a specified number of digits after the decimal point—up to a maximum of 32 million. It uses Gauss–Legendre algorithm and is a Windows port of the program used by Yasumasa Kanada in 1995 to compute pi to 2 32 digits.