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In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. [1] [2] It is denoted by π(x) (unrelated to the number π). A symmetric variant seen sometimes is π 0 (x), which is equal to π(x) − 1 ⁄ 2 if x is exactly a prime number, and equal to π(x) otherwise ...
The first such distribution found is π(N) ~ N / log(N) , where π(N) is the prime-counting function (the number of primes less than or equal to N) 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 ) .
There are known formulae to evaluate the prime-counting function (the number of primes smaller than a given value) faster than computing the primes. This has been used to compute that there are 1,925,320,391,606,803,968,923 primes (roughly 2 × 10 21 ) smaller than 10 23 .
The least count is related to the precision of an instrument; an instrument that can measure smaller changes in a value relative to another instrument, has a smaller "least count" value and so is more precise. Any measurement made by the instrument can be considered repeatable to no less than the resolution of the least count.
Presumably from the practice, in counting sheep or large herds of cattle, of counting orally from one to twenty, and making a score or notch on a stick, before proceeding to count the next twenty. [3] [4] A distance of twenty yards in ancient archery and gunnery. [5] Threescore: 60 Three score (3x20) Large: 1,000 Slang for one thousand Myriad ...
In the graph at right the top line y = n − 1 is an upper bound valid for all n other than one, and attained if and only if n is a prime number. A simple lower bound is φ ( n ) ≥ n / 2 {\displaystyle \varphi (n)\geq {\sqrt {n/2}}} , which is rather loose: in fact, the lower limit of the graph is proportional to n / log log n .
Similarly, the totient is equal to 4 when n is one of the four values 5, 8, 10, and 12, and it is equal to 6 when n is one of the four values 7, 9, 14, and 18. In each case, there is more than one value of n having the same value of φ(n). The conjecture states that this phenomenon of repeated values holds for every n.
The high-water marks for occur for n = 1, 2, and 4, with A 4 ≈ 0.670873..., with no larger value among the first 10 5 primes. Since the Andrica function decreases asymptotically as n increases, a prime gap of ever increasing size is needed to make the difference large as n becomes large. It therefore seems highly likely the conjecture is true ...