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Introduced in Python 2.2 as an optional feature and finalized in version 2.3, generators are Python's mechanism for lazy evaluation of a function that would otherwise return a space-prohibitive or computationally intensive list. This is an example to lazily generate the prime numbers:
Therefore, every prime number other than 2 is an odd number, and is called an odd prime. [10] Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are even, and decimal numbers that end in ...
It can mean just "not provable", leaving open whether an independent statement might be refuted. Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of whether the truth value of the statement is well-defined, or whether it can be determined by other means.
A classic example of a decidable decision problem is the set of prime numbers. It is possible to effectively decide whether a given natural number is prime by testing every possible nontrivial factor. Although much more efficient methods of primality testing are known, the existence of any effective method is enough to establish decidability.
The phenomenon that π 4,3 (x) is ahead most of the time is called Chebyshev's bias. The prime number race generalizes to other moduli and is the subject of much research; Pál Turán asked whether it is always the case that π c,a (x) and π c,b (x) change places when a and b are coprime to c. [34] Granville and Martin give a thorough ...
Since 2 divides , +, and +, and 3 divides and +, the only possible remainders mod 6 for a prime greater than 3 are 1 and 5. So, a more efficient primality test for n {\displaystyle n} is to test whether n {\displaystyle n} is divisible by 2 or 3, then to check through all numbers of the form 6 k + 1 {\displaystyle 6k+1} and 6 k + 5 ...
In Python 3.x the range() function [28] returns a generator which computes elements of the list on demand. Elements are only generated when they are needed (e.g., when print(r[3]) is evaluated in the following example), so this is an example of lazy or deferred evaluation:
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