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... Note that given an escalator number A = An , for each m > n, the values of am and Am are completely determined. We now investigate the degree to which these values are determined for m ≤ n. Let a1 , a2 , a3 , a4 , . . . be an escalator sequence. Since A2 = a1 + a2 = a1 a2 , it’s also true that A2 = ...
... Note that given an escalator number A = An , for each m > n, the values of am and Am are completely determined. We now investigate the degree to which these values are determined for m ≤ n. Let a1 , a2 , a3 , a4 , . . . be an escalator sequence. Since A2 = a1 + a2 = a1 a2 , it’s also true that A2 = ...
Floating point numbers in Scilab
... x ∈ R for which there exists at least one representation (M, e) such that the equation 3 holds. By at least, we mean that it might happen that the real number x is either too large or too small. In this case, no couple (M, e) can be found to satisfy the equations 3, 4 and 5. This point will be revie ...
... x ∈ R for which there exists at least one representation (M, e) such that the equation 3 holds. By at least, we mean that it might happen that the real number x is either too large or too small. In this case, no couple (M, e) can be found to satisfy the equations 3, 4 and 5. This point will be revie ...
4-RSA
... The order of G, ord(G ), is the number of elements in G. The order of a G, written ord( a ), is the smallest positive integer t such that a t e. (e, identity element.) Lagrange's theorem: For any element a G, ord( a ) | ord(G ). Corollary: For any element a G, a ord( G ) e. Ferma ...
... The order of G, ord(G ), is the number of elements in G. The order of a G, written ord( a ), is the smallest positive integer t such that a t e. (e, identity element.) Lagrange's theorem: For any element a G, ord( a ) | ord(G ). Corollary: For any element a G, a ord( G ) e. Ferma ...
Fibonacci numbers
... numbers converges. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that the limit approaches the golden ratio .[20] ...
... numbers converges. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that the limit approaches the golden ratio .[20] ...
A first introduction to p-adic numbers
... above. To inverse a p-adic integer α ending in a digit d other than 0 and 1, we find the (unique) digit f such that df is congruent to 1 mod p (i.e. is equal to 1 plus a multiple of p). In that case, f α ends in 1 so can be inverted, and we then have 1/α = f /(f α). To find f for small values of p, ...
... above. To inverse a p-adic integer α ending in a digit d other than 0 and 1, we find the (unique) digit f such that df is congruent to 1 mod p (i.e. is equal to 1 plus a multiple of p). In that case, f α ends in 1 so can be inverted, and we then have 1/α = f /(f α). To find f for small values of p, ...
An amazing prime heuristic
... or Cullen numbers suggest that there are infinitely many primes of each of these forms. But it would also imply there are infinitely many primes of the form 3 n − 1, even though all but one of these are composite. So we must be a more careful than just adding up the terms 1/ log n. We will illustrat ...
... or Cullen numbers suggest that there are infinitely many primes of each of these forms. But it would also imply there are infinitely many primes of the form 3 n − 1, even though all but one of these are composite. So we must be a more careful than just adding up the terms 1/ log n. We will illustrat ...
Variables, Expressions and Statements
... Write a script that prints X and Y in order, then exchange their values (X becomes 'Dog' and Y 'Cat') and prints X and Y again. Cat Dog Dog Cat ...
... Write a script that prints X and Y in order, then exchange their values (X becomes 'Dog' and Y 'Cat') and prints X and Y again. Cat Dog Dog Cat ...
Limits of Functions
... 1. If possible, use the quotient theorem for limits. 2. If lim f(x) = 0 and lim g(x) = 0, try the following techniques. f x a. Factor g x and f x and reduce to lowest terms. g x b. If f x or g x involves a square root, try multiplying both ...
... 1. If possible, use the quotient theorem for limits. 2. If lim f(x) = 0 and lim g(x) = 0, try the following techniques. f x a. Factor g x and f x and reduce to lowest terms. g x b. If f x or g x involves a square root, try multiplying both ...
31(1)
... at a perfect phi pyramid. Maybe the architect's plans will eventually be found entombed with his mummy. ...
... at a perfect phi pyramid. Maybe the architect's plans will eventually be found entombed with his mummy. ...
Law of large numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.The LLN is important because it ""guarantees"" stable long-term results for the averages of some random events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. It is important to remember that the LLN only applies (as the name indicates) when a large number of observations are considered. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be ""balanced"" by the others (see the gambler's fallacy)