如何随机生成一个数字，其条件与其他两个随机生成的数字没有共同因素

Question

p1 = 1097
p2 = 1279
p = p1*p2
phi = (p1-1)*(p1-1)

I want to randomly generate a number >phi that does not share common factors between "p" and "phi"

Answer 1

Constructing the numbers from known primes so they do not share the same primes (as factors) in them should be faster then creating random numbers, factorizing them and checking them for non-overlapping factors.

Algo:

• calculate/provide enough primes to work with (that start with your lower bound)
• then create 2 lists of random indexes into the primes list
  • first list draw how many numbers you want as factors
  • remove any indexes from the 2nd list that are already in the 1st list until both lists confirm to your wanted number of factors
• then multpiply the primes together

could be done like so:

# get or calculate some primes that are big enough
# starting with 
primes = [1097,1103,1109,1117,1123,1129,1151,
  1153,1163,1171,1181,1187,1193,1201,1213,1217,1223,1229,1231,
  1237,1249,1259,1277,1279,1283,1289,1291,1297,1301,1303,1307,
  1319,1321,1327,1361,1367,1373,1381,1399,1409,1423,1427,1429,
  1433,1439,1447,1451,1453,1459,1471,1481,1483,1487,1489,1493,
  1499,1511,1523,1531,1543,1549,1553,1559,1567,1571,1579,1583,
  1597,1601,1607,1609,1613,1619,1621,1627,1637,1657,1663,1667,
  1669,1693,1697,1699,1709,1721,1723,1733,1741,1747,1753,1759,
  1777,1783,1787,1789,1801,1811,1823,1831,1847,1861,1867,1871,
  1873,1877,1879,1889,1901,1907,1913,1931,1933,1949,1951,1973,
  1979,1987,1993,1997,1999,2003,2011,2017,2027,2029,2039,2053,
  2063,2069,2081,2083,2087,2089,2099,2111,2113,2129,2131,2137,
  2141,2143,2153,2161,2179,2203,2207,2213,2221,2237,2239,2243,
  2251,2267,2269,2273,2281,2287]

then choose indexes into them and multiply them to get your result:

import random
from operator import mul
from functools import reduce

random.seed(42)
choices = random.choices

def m(numbers):
    return reduce(mul, numbers, 1)

# how many numbers to use to factorize the number you want    
size = 5

# draw f.e. 5 factors from then
p1 = [primes[i] for i in choices(range(len(primes)), k=size)]

# draw 5 others that are not in p1
p2 = []
while len(p2) != size:
  p2 = [primes[i] for i in choices(range(len(primes)), k=3 * size) if i not in p1][:size]

# calculate them 
print( m(p1), m(p2)) # 8058608274530453 8642866553052643

Your choice of primes to be used should be bigger then twice the numbers you want to use to multiply out your results else you might hang in an long running loop. Providing about 5 to 10 times the primes that you want to use as factors should be plenty.

Answer 2

Not sure what you mean by "don't sharing common factors between p and phi", as p and phi won't have common factors

One of the "bruteforce" ways to do it would be to run a Poisson random variable to get an integer n, then to compute the decomposition of each integer bigger than psi until you get the n-th integer bigger than psi satisfying your condition.

Another way I would see would be to take a random integer bigger than psi (with for instance a Poisson random variable) and check if they satisfy your condition.

Depends on what you mean and the size of your numbers.

EDIT: I suggest Poisson variables as they allow you to avoid setting an arbitrary hardcoded upper bound for your random number

Answer 3

You can generate random numbers using random module in python.

To find that there is no common factors with p and phi , you can check greatest common divisor (GCD) with p*phi .

from math import gcd as calcGcd
from random import randint

def getRandom(lower, upper, *noFacWithThese):
    for _ in range(upper - lower): # to avoid infinite loop
        randNum = randint(lower, upper)
        if noCommonFactors(randNum, noFacWithThese):
            return randNum
    return None

def noCommonFactors(base, withThese):
    withThis = 1
    for n in withThese: # find mutiplication of numbers inside 'withThese' list
        withThis *= n
    return calcGcd(base, withThis) == 1

p1 = 1097
p2 = 1279
p = p1*p2
phi = (p1-1)*(p1-1)

while True:
    print( getRandom(1, 10000, p, phi) )
    input()
    

Make sure to use necessary lower and upper limits for randint function.