Initial data commit

Task/Multiplicative-order/Python/multiplicative-order.py

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+def gcd(a, b):
+    while b != 0:
+        a, b = b, a % b
+    return a
+
+def lcm(a, b):
+    return (a*b) / gcd(a, b)
+
+def isPrime(p):
+    return (p > 1) and all(f == p for f,e in factored(p))
+
+primeList = [2,3,5,7]
+def primes():
+    for p in primeList:
+        yield p
+    while 1:
+        p += 2
+        while not isPrime(p):
+            p += 2
+        primeList.append(p)
+        yield p
+
+def factored( a):
+    for p in primes():
+        j = 0
+        while a%p == 0:
+            a /= p
+            j += 1
+        if j > 0:
+            yield (p,j)
+        if a < p*p: break
+    if a > 1:
+        yield (a,1)
+
+
+def multOrdr1(a,(p,e) ):
+    m = p**e
+    t = (p-1)*(p**(e-1)) #  = Phi(p**e) where p prime
+    qs = [1,]
+    for f in factored(t):
+        qs = [ q * f[0]**j for j in range(1+f[1]) for q in qs ]
+    qs.sort()
+
+    for q in qs:
+        if pow( a, q, m )==1: break
+    return q
+
+
+def multOrder(a,m):
+    assert gcd(a,m) == 1
+    mofs = (multOrdr1(a,r) for r in factored(m))
+    return reduce(lcm, mofs, 1)
+
+
+if __name__ == "__main__":
+    print multOrder(37, 1000)        # 100
+    b = 10**20-1
+    print multOrder(2, b) # 3748806900
+    print multOrder(17,b) # 1499522760
+    b = 100001
+    print multOrder(54,b)
+    print pow( 54, multOrder(54,b),b)
+    if any( (1==pow(54,r, b)) for r in range(1,multOrder(54,b))):
+        print 'Exists a power r < 9090 where pow(54,r,b)==1'
+    else:
+        print 'Everything checks.'