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Adding faster method using a reduced version of difference of squares I made up (but may or may not have already been dreamed up by someone else)

Author: GarrickLarson

Reduced_Difference_of_Squares.py

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+import math
+import time
+# from sympy import nextprime
+from decimal import *
+# Garrick Larson, Spring 2018
+getcontext().prec = 1250 # precision of square root function. lower precision = higher speed, higher precision = higher prime factors
+
+# 1200 digits
+factor1 = 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000005813
+factor2 = 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000005227
+
+# factor1 = 191
+# factor2 = 103
+# factor1 = 86028157
+# factor2 = 90713509
+
+sprime = Decimal((factor1*factor2))
+getcontext().prec = math.ceil(Decimal(sprime).log10()*Decimal(1.1)) # precision of square root function. lower precision = higher speed, higher precision = higher prime factors
+
+start_time = time.time()
+
+print()
+print('Factor 1: ')
+print(factor1)
+print()
+print('Factor 2: ')
+print(factor2)
+print()
+print('Composite Semiprime: ')
+print(sprime)
+print()
+
+sq = math.ceil(Decimal(sprime).sqrt())
+remainder_orig = sq**2 - sprime
+sq2 = sq*2 + 1
+remainder = remainder_orig
+# print(sq2)
+# print(remainder)
+count = 1
+
+r = Decimal(remainder).sqrt()
+while r != (r//1):
+    remainder += sq2
+    sq2 += 2
+    count += 1
+    if count % 1000 == 0:
+        print(count)
+    r = Decimal(remainder).sqrt()
+
+# print("remainder: " + str(remainder) + "\n")
+# print("r: " + str(r) + "\n")
+# print("sq2: " + str(sq2) + "\n")
+
+
+
+difference = r
+factor1_output = Decimal(sprime+difference**2).sqrt() + difference
+factor2_output = Decimal(sprime+difference**2).sqrt() - difference
+
+print()
+print("Factor 1 Output:")
+print(factor1_output)
+print()
+print("Factor 2 Output:")
+print(factor2_output)
+print()
+print("Composite: ")
+print(factor1_output*factor2_output)
+print()
+print("count: " + str(count) + "\n")
+
+print("--- %s seconds to run   ---" % (time.time() - start_time))