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- #!/usr/bin/ruby
- # Daniel "Trizen" Șuteu
- # Date: 03 June 2019
- # https://github.com/trizen
- # A simple factorization method for numbers close to a perfect power.
- # Very effective for numbers of the form:
- #
- # n^k - 1
- #
- # where k has many divisors.
- func near_power_factorization(n, bound=100) {
- var orig = n
- func f(r, e, k) {
- var factors = gather {
- e.divisors.each {|d|
- for j in (1, -1) {
- var t = (r**d - k*j)
- var g = gcd(t, n)
- if (g.is_between(2, n-1)) {
- while (g.divides(n)) {
- n /= g
- take(g)
- }
- }
- }
- }
- }
- factors << orig/factors.prod
- factors.sort
- }
- for j in (1..bound) {
- for k in (1, -1) {
- var u = (k * j**2)
- if (is_power(n + u)) {
- var r = perfect_root(n + u)
- var e = perfect_power(n + u)
- return f(r, e, j)
- }
- }
- }
- return [n]
- }
- if (ARGV) {
- say near_power_factorization(Num(ARGV[0]))
- return 1
- }
- say near_power_factorization(2**256 - 1) #=> [3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457]
- say near_power_factorization(10**120 + 1) #=> [100000001, 9999999900000001, 99999999000000009999999900000001, 10000000099999999999999989999999899999999000000000000000100000001]
- say near_power_factorization(10**120 - 1) #=> [3, 9, 11, 37, 91, 101, 9091, 9901, 10001, 11111, 90090991, 99009901, 99990001, 109889011, 9999000099990001, 10099989899000101, 100009999999899989999000000010001]
- say near_power_factorization(10**120 - 25) #=> [3, 5, 5, 29, 2298850574712643678160919540229885057471264367816091954023, 199999999999999999999999999999999999999999999999999999999999]
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