carmichael_generation_erdos_method_dynamic_programming.sf

#!/usr/bin/ruby

# Three approaches at implementing the Erdos method for constructing Carmichael numbers, using dynamic programming.

# Erdos construction method for Carmichael numbers:
#   1. Choose an integer L with many prime factors.
#   2. Let P be the set of primes d+1, where d|L and d+1 does not divide L.
#   3. Find a subset S of P such that prod(S) == 1 (mod L). Then prod(S) is a Carmichael number.

# Alternatively:
#   3. Find a subset S of P such that prod(S) == prod(P) (mod L). Then prod(P) / prod(S) is a Carmichael number.

func carmichael_numbers_from_lambda(L, callback) {    # smallest numbers first

    var P = L.divisors.map { .inc }.grep {|p| p.is_odd && p.is_prime && L.is_coprime(p) }

    var d = [1]

    P.each {|p|
        d << gather {
            d.each { |u|
                take(var c = u*p)
                if (c.is_congruent(1, L)) {
                    callback(c)
                }
            }
        }...
    }

    return nil
}

func carmichael_numbers_from_lambda_large(L, callback) {  # largest numbers first

    var P = L.divisors.map { .inc }.grep {|p| p.is_odd && p.is_prime && L.is_coprime(p) }

    var T = P.prod
    var s = T%L
    var d = [1]

    P.each {|p|
        d << gather {
            d.each { |u|
                take(var c = u*p)
                if (c.is_congruent(s, L)) {
                    callback(idiv(T, c)) if (T > c)
                }
            }
        }...
    }

    return nil
}

func carmichael_numbers_from_lambda_lazy(L, callback) {    # on-demand version

    var d1 = [1]
    var d2 = [1]

    for p,e in (L.factor_exp) {
        var r = 1
        d1 << gather {
            e.times {
                r *= p
                d1.each { |u|
                    take(var t = u*r)

                    var q = t+1
                    next if !(q.is_odd && q.is_prime && L.is_coprime(q))

                    d2 << gather {
                        d2.each { |u|
                            take(var c = u*q)
                            if (c.is_congruent(1, L)) {
                                callback(c)
                            }
                        }
                    }...
                }
            }
        }...
    }

    return nil
}

carmichael_numbers_from_lambda(720, {|c| say c })
carmichael_numbers_from_lambda_lazy(720, {|c| say c })
carmichael_numbers_from_lambda_large(720, {|c| say c })