Full text
... Proof (and construction) : Suppose the value of the least term, in magnitude, that repeats infinitely often is k. If there is no term of f greater than k, then the terms of / less than k can be reordered, and the term k (infinitely repeated) tacked on to the end, to obtain the sequence g. By this pr ...
... Proof (and construction) : Suppose the value of the least term, in magnitude, that repeats infinitely often is k. If there is no term of f greater than k, then the terms of / less than k can be reordered, and the term k (infinitely repeated) tacked on to the end, to obtain the sequence g. By this pr ...
Alg-supp
... Suggest you work out some cases As many cases as it takes… © Leong Hon Wai, 2003-2008 LeongHW, SoC, NUS ...
... Suggest you work out some cases As many cases as it takes… © Leong Hon Wai, 2003-2008 LeongHW, SoC, NUS ...
Mathematical Investigation: Paper Size
... natural numbers?” All odd numbers, except 1, can be expressed as the sum of consecutive natural numbers (see Q1-4). The problem is the even numbers. All even natural numbers can be classified into two categories: multiples of odd numbers (excluding multiples of 1) and powers of 2. If you divid ...
... natural numbers?” All odd numbers, except 1, can be expressed as the sum of consecutive natural numbers (see Q1-4). The problem is the even numbers. All even natural numbers can be classified into two categories: multiples of odd numbers (excluding multiples of 1) and powers of 2. If you divid ...
34(3)
... dividing m' and that qR is the smallest prime not dividing mf. It also follows that ft_M+1 > qR, since m' would be simple otherwise. Hence p£__u > qR+l, because m is not simple. The proof when m is a left child is similar. D Fix a path from n to $j. A left child has one less prime divisor than its p ...
... dividing m' and that qR is the smallest prime not dividing mf. It also follows that ft_M+1 > qR, since m' would be simple otherwise. Hence p£__u > qR+l, because m is not simple. The proof when m is a left child is similar. D Fix a path from n to $j. A left child has one less prime divisor than its p ...
A Study to the 3n+1 Problem with State Transition Model
... makes it qualified. For example, giving 10101 as n, the result of 3n is 101010+10101=111111, and that of 3n+1 is 1000000. It is trivial that if n is qualified then 2n is qualified, too. Hence, we have to pay attention to odd numbers only. Binary numbers 11, 111, 1111, 11111 …, which are with more th ...
... makes it qualified. For example, giving 10101 as n, the result of 3n is 101010+10101=111111, and that of 3n+1 is 1000000. It is trivial that if n is qualified then 2n is qualified, too. Hence, we have to pay attention to odd numbers only. Binary numbers 11, 111, 1111, 11111 …, which are with more th ...
Public-Key Crypto Basics Paul Garrett
... Why is it feasible for Alice to find two primes ∼ 10200 or so? Why is it feasible for Alice to compute e−1 mod (p − 1)(q − 1)? Why is it feasible for Bob to compute xe mod n? Why is it feasible for Alice to compute y d mod n? Why is y d mod n = x? Why is it not feasible for Eve (the eavesdropper) to ...
... Why is it feasible for Alice to find two primes ∼ 10200 or so? Why is it feasible for Alice to compute e−1 mod (p − 1)(q − 1)? Why is it feasible for Bob to compute xe mod n? Why is it feasible for Alice to compute y d mod n? Why is y d mod n = x? Why is it not feasible for Eve (the eavesdropper) to ...
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS
... the positive number . If 0 is a smaller positive number than , then the corresponding N 0 very likely will need to be larger than N. Sometimes we will indicate this dependence by writing N () instead of simply N. It is always wise to remember that N depends on . On the other hand, the N or N ( ...
... the positive number . If 0 is a smaller positive number than , then the corresponding N 0 very likely will need to be larger than N. Sometimes we will indicate this dependence by writing N () instead of simply N. It is always wise to remember that N depends on . On the other hand, the N or N ( ...
rational number
... denominator denominator numerator When writing a long division problem from a fraction, put the numerator inside the “box,” or division symbol. It may help to write the numerator first and then say “divided by” to yourself as you write the ...
... denominator denominator numerator When writing a long division problem from a fraction, put the numerator inside the “box,” or division symbol. It may help to write the numerator first and then say “divided by” to yourself as you write the ...
02-Signed Number Systems
... Consider the fractional part of the value in d Fixed point system X F x1 d 1 x2 d 2 ...
... Consider the fractional part of the value in d Fixed point system X F x1 d 1 x2 d 2 ...
SEVEN CONSECUTIVE PRIMES IN ARITHMETIC PROGRESSION
... 22 [6]. A related conjecture is the following: there exist arbitrarily long sequences of consecutive primes in arithmetic progression [2]. In 1967, Lander and Parkin [4] reported finding the first and smallest sequence of 6 consecutive primes in AP, where the starting prime is 121174811 and the comm ...
... 22 [6]. A related conjecture is the following: there exist arbitrarily long sequences of consecutive primes in arithmetic progression [2]. In 1967, Lander and Parkin [4] reported finding the first and smallest sequence of 6 consecutive primes in AP, where the starting prime is 121174811 and the comm ...
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)