100011322432435545
May 18, 2021
I had an email from a reader who found my factoring program on Stack Overflow:
Python 2.7.13 (default, Mar 13 2017, 20:56:15) [GCC 5.4.0] on cygwin Type "help", "copyright", "credits" or "license" for more information. >>> def isPrime(n, k=5): # miller-rabin ... from random import randint ... if n < 2: return False ... for p in [2,3,5,7,11,13,17,19,23,29]: ... if n % p == 0: return n == p ... s, d = 0, n-1 ... while d % 2 == 0: ... s, d = s+1, d/2 ... for i in range(k): ... x = pow(randint(2, n-1), d, n) ... if x == 1 or x == n-1: continue ... for r in range(1, s): ... x = (x * x) % n ... if x == 1: return False ... if x == n-1: break ... else: return False ... return True ... >>> def factors(n, b2=-1, b1=10000): # 2,3,5-wheel, then rho ... def gcd(a,b): # euclid's algorithm ... if b == 0: return a ... return gcd(b, a%b) ... def insertSorted(x, xs): # linear search ... i, ln = 0, len(xs) ... while i < ln and xs[i] < x: i += 1 ... xs.insert(i,x) ... return xs ... if -1 <= n <= 1: return [n] ... if n < -1: return [-1] + factors(-n) ... wheel = [1,2,2,4,2,4,2,4,6,2,6] ... w, f, fs = 0, 2, [] ... while f*f <= n and f < b1: ... while n % f == 0: ... fs.append(f) ... n /= f ... f, w = f + wheel[w], w+1 ... if w == 11: w = 3 ... if n == 1: return fs ... h, t, g, c = 1, 1, 1, 1 ... while not isPrime(n): ... while b2 <> 0 and g == 1: ... h = (h*h+c)%n # the hare runs ... h = (h*h+c)%n # twice as fast ... t = (t*t+c)%n # as the tortoise ... g = gcd(t-h, n); b2 -= 1 ... if b2 == 0: return fs ... if isPrime(g): ... while n % g == 0: ... fs = insertSorted(g, fs) ... n /= g ... h, t, g, c = 1, 1, 1, c+1 ... return insertSorted(n, fs)
He complained that the program failed to factor the number 100011322432435545, entering an infinite loop instead of returning a list of factors.
Your task is to figure out what is wrong, and fix it. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.
Potholes
May 11, 2021
Today’s exercise comes from one of those online code exercises, via Stack Overflow:
There is a machine that can fix all potholes along a road 3 units in length. A unit of road will be represented by a period in a string. For example, “…” is one section of road 3 units in length. Potholes are marked with an “X” in the road, and also count as 1 unit of length. The task is to take a road of length N and fix all potholes with the fewest possible sections repaired by the machine.
The city where I live needs one of those machines.
Here are some examples:
.X. 1 .X...X 2 XXX.XXXX 3 .X.XX.XX.X 3
Your task is to write a program that takes a string representing a road and returns the smallest number of fixes that will remove all the potholes. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.