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A necessary (but not sufficient) condition for solvability is that n is not divisible by 4 or by a prime of form 4k + 3. [note 3] Thus, for example, x 2 − 3 y 2 = −1 is never solvable, but x 2 − 5 y 2 = −1 may be. [27] The first few numbers n for which x 2 − n y 2 = −1 is solvable are with only one trivial solution: 1
where f (2k−1) is the (2k − 1)th derivative of f and B 2k is the (2k)th Bernoulli number: B 2 = 1 / 6 , B 4 = − + 1 / 30 , and so on. Setting f ( x ) = x , the first derivative of f is 1, and every other term vanishes, so [ 15 ]
Simplification is the process of replacing a mathematical expression by an equivalent one that is simpler (usually shorter), according to a well-founded ordering. Examples include:
49 is the smallest discriminant of a totally real cubic field. [2] 49 and 94 are the only numbers below 100 whose all permutations are composites but they are not multiples of 3, repdigits or numbers which only have digits 0, 2, 4, 5, 6 and 8, even excluding the trivial one digit terms. 49 = 7^2 and 94 = 2 * 47
The TI-108 is a simple four-function calculator which uses single-step execution.. The immediate execution mode of operation (also known as single-step, algebraic entry system (AES) [7] or chain calculation mode) is commonly employed on most general-purpose calculators.
[8] 1857 Panic of 1857; NJH&D can't get together enough money and sells its charter to the Pennsylvania Coal Company. [3] [9] 1866 DeWitt Clinton Littlejohn forms the New York and Oswego Midland Railway 400 miles to the west of New York City to connect Oswego, New York with the port of New York [3] Hoboken Ridgefield & Paterson chartered on ...
An orange that has been sliced into two halves. In mathematics, division by two or halving has also been called mediation or dimidiation. [1] The treatment of this as a different operation from multiplication and division by other numbers goes back to the ancient Egyptians, whose multiplication algorithm used division by two as one of its fundamental steps. [2]
90 is the only number to have an aliquot sum of 144 = 12 2. Only three numbers have a set of divisors that generate a sum equal to 90, they are 40, 58, and 89. [3] 90 is also the twentieth abundant [4] and highly abundant [5] number (with 20 the first primitive abundant number and 70 the second). [6] The number of divisors of 90 is 12. [7]