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It always answers true for prime number inputs; for composite inputs, it answers false with probability at least 1 ⁄ 2 and true with probability less than 1 ⁄ 2. Thus, false answers from the algorithm are certain to be correct, whereas the true answers remain uncertain; this is said to be a 1 ⁄ 2-correct false-biased algorithm.
If n is odd, the binary OR operation n|1 has no effect, so this is equivalent to t = n * ((n+1)/2) and thus correct. If n is even, setting the low bit with n|1 is the same as adding 1, while the 1 added before the division is truncated away, so this is equivalent to t = (n+1) * (n/2) and also correct.
Lucas numbers have L 1 = 1, L 2 = 3, and L n = L n−1 + L n−2. Primefree sequences use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite. Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The Pell numbers have P n = 2P n−1 + P n−2.
[note 1] Another classical example of a howler is proving the Cayley–Hamilton theorem by simply substituting the scalar variables of the characteristic polynomial with the matrix. Bogus proofs, calculations, or derivations constructed to produce a correct result in spite of incorrect logic or operations were termed "howlers" by Edwin Maxwell. [2]
Alternatively, and for greater numbers, one may say for 1 ⁄ 2 "one over two", for 5 ⁄ 8 "five over eight", and so on. This "over" form is also widely used in mathematics. Fractions together with an integer are read as follows: 1 + 1 ⁄ 2 is "one and a half" 6 + 1 ⁄ 4 is "six and a quarter" 7 + 5 ⁄ 8 is "seven and five eighths"
In mathematics, the infinite series 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1. In summation notation, this may be expressed as
1 is read off as "one 1" or 11. 11 is read off as "two 1s" or 21. 21 is read off as "one 2, one 1" or 1211. 1211 is read off as "one 1, one 2, two 1s" or 111221. 111221 is read off as "three 1s, two 2s, one 1" or 312211. The look-and-say sequence was analyzed by John Conway [1] after he was introduced to it by one of his students at a party. [2 ...
The line segment ¯ has length and sum of the lengths of ¯ and ¯ equals the length of ¯, which is 1. Therefore, cos 2 θ + 2 sin 2 θ = 1 {\displaystyle \cos 2\theta +2\sin ^{2}\theta =1} .