Project 1: list comprehensions
... CS 5035 (Fall 2016) Project 1. Introduction and starting out (first attempt by Aug 29) Based on chapters 1 and 2 of LYH. Videos. Use list comprehension to generate the 2 non-prime odd numbers less than 6,000 not of the form p + 2 * k^2, where p is prime and k>0 . (The answer: [5777, 5993].) This is ...
... CS 5035 (Fall 2016) Project 1. Introduction and starting out (first attempt by Aug 29) Based on chapters 1 and 2 of LYH. Videos. Use list comprehension to generate the 2 non-prime odd numbers less than 6,000 not of the form p + 2 * k^2, where p is prime and k>0 . (The answer: [5777, 5993].) This is ...
n=1
... “One would of course like to have a rigorous proof of this, but I have put aside the search for such a proof after some fleeting vain attempts because it is not necessary for the immediate objective of my investigation” ...
... “One would of course like to have a rigorous proof of this, but I have put aside the search for such a proof after some fleeting vain attempts because it is not necessary for the immediate objective of my investigation” ...
FURTHER PROPERTIES OF THE NUMBER FRACTION FUNCTION
... many divisors. One also sees that the function vanishes when N is a prime. We call these primes the Q Primes. They represent essentially all primes above p=3 and are shown in blue at the bottom of the graph. They can all be represented by 6n±1. Semi-Primes N=pq typically are found to have very small ...
... many divisors. One also sees that the function vanishes when N is a prime. We call these primes the Q Primes. They represent essentially all primes above p=3 and are shown in blue at the bottom of the graph. They can all be represented by 6n±1. Semi-Primes N=pq typically are found to have very small ...
Prime number theorem
In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée-Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function).The first such distribution found is π(N) ~ N / log(N), where π(N) is the prime-counting function and log(N) is the natural logarithm of N. This means that for large enough N, the probability that a random integer not greater than N is prime is very close to 1 / log(N). Consequently, a random integer with at most 2n digits (for large enough n) is about half as likely to be prime as a random integer with at most n digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (log(101000) ≈ 2302.6), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (log(102000) ≈ 4605.2). In other words, the average gap between consecutive prime numbers among the first N integers is roughly log(N).