Vocabulary/pco
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p: y Primes
Rank 0 -- operates on individual atoms of y, producing a result of the same shape -- WHY IS THIS IMPORTANT?
The y-th prime (starting with 2 as the 0-th prime).
p: i.9 2 3 5 7 11 13 17 19 23 p: 2000 17393
The inverse p:^:_1 y tells the number of primes less than y
p:^:_1 (17 19 23 24 25) 6 7 8 9 9
Common uses
Mathematical investigations.
Related Primitives
Prime Factors * Prime Exponents (q:)
x p: y | Primes |
Rank Infinity -- operates on x and y as a whole -- WHY IS THIS IMPORTANT?
A collection of prime-related functions of integer y, with x selecting the function.
The Prime Functions x p: y | |
x | Function |
_4 | The largest prime smaller than y |
_1 | π(y), the number of primes less than y (same as p:^:_1) |
0 | 1 if y is not prime |
1 | 1 if y is prime |
2 | a 2-row table of the prime factors and exponents in the factorization of y (same as __ q: y) |
3 | the list of primes whose product is equal to y (same as q: y) |
4 | The smallest prime larger than y |
5 | The number of integers less than or equal to y that are relatively prime to y (Euler's totient function φ(y)) |
]y=: p: 2001 NB. The 2001st prime 17401 _4 p: y NB. Next prime down from y 17393 4 p: 17393 NB. Next prime up from 17393 17401 _1 p: y NB. Number of primes <y 2001 1 p: y NB. Boolean: y is prime 1 0 p: y NB. Boolean: y is non-prime 0 1 p: i.25 NB. flags the primes in i.25 0 0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 I. (1 p: i.25) NB. Primes in (i.25) 2 3 5 7 11 13 17 19 23 3 p: 2+ 2^31 NB. prime factorization of 2+(2^31) 2 5 5 13 41 61 1321 2 p: 2+ 2^31 NB. table of prime factors of 2+(2^31) with their exponents 2 5 13 41 61 1321 1 2 1 1 1 1 NB. Totient function φ(y) = Number of co-prime integers less than or equal to y NB. φ(y) = y-1 (if y is prime) and less of course if y is composite 5 p: y NB. y (17401) is prime 17400 3 p: y-1 NB. y-1 (17400) is composite (non-prime) 2 2 2 3 5 5 29 2 p: y-1 NB. showing distinct prime factors 2 3 5 29 3 1 2 1 NB. using Product over the distinct prime factors (y-1)*(1-1r2)*(1-1r3)*(1-1r5)*(1-1r29) 4480 5 p: y-1 NB. using Totient function shortcut 4480
Common uses
Mathematical investigations.
Related Primitives
Prime Factors * Prime Exponents (q:)
Details
1. Primality testing on numbers larger than (2^31) uses the probabilistic Miller-Rabin algorithm.