Prime Numbers and Goldbach's Conjecture with Python

Prime numbers (those integers that cannot be divided neatly without fractions or remainders by any other numbers except 1 and itself) are big business. As in they are used in business mainly for secure transactions. Other folks just find them interesting in their own right, speculating whether they follow a predictable pattern (which we have not discovered yet) or not.

The method I have written for finding them is not nearly as sophisticated as those of more skilled mathematicians. It uses a method called the Sieve of Eratosthenes. Basically the program takes a number and tries to divide it by all the numbers below it (except for 1). If any do divide neatly, then it is not a prime number.

Another mathematician called Goldbach came up with a conjecture that any non-prime number could be created by adding two prime numbers together. So for example, 2 is a prime, 4 is not a prime and can be expressed as 2 * 2, or in Goldbach's conjecture 2+2.
Goldbach's conjecture seems to hold true as far as practical calculations show, but nobody is sure if that is because of some important underlying principle or if it is just very easy to do so.

So this program asks for a maximum number. It then goes from 2 to the maximum number, using the sieve of Eratosthenes to find if each number is a prime, and if not then whether it fits Goldbach's conjecture.

#!/usr/bin/python3

def goldbach(evennumber):
    for x in primelist:
        for y in primelist:
            if x + y == evennumber:
                print(", Goldbach conjecture: "+str(x)+" + "+str(y)+" = "+str(evennumber))
                return()
    print("Problem with Goldbach Conjecture!"); exit()

count = 0
maxcount = input("Please enter maximum count: ")
try: int(maxcount)
except: print("Bad input"); exit()
else: maxcount = int(maxcount)
testnumber = 2
primelist = [2]
while count < maxcount-2 :
    testnumber +=1
    prime = True
    count += 1
    for testfactor in primelist:
        if testnumber % testfactor == 0:
            print(str(testnumber)+" is divisible by "+str(testfactor), end='')
            prime = False
            if testfactor == 2: goldbach(testnumber)
            else: print('')
            break
    if prime == True:
        primelist.append(testnumber)
        print(str(testnumber) + " is a prime!")

#print(primelist)
print("Length of prime list is "+str(len(primelist)))

And a typical output is:

>>> ================================ RESTART ================================
>>>
Please enter maximum count: 20003 is a prime!
4 is divisible by 2, Goldbach conjecture: 2 + 2 = 4
5 is a prime!
6 is divisible by 2, Goldbach conjecture: 3 + 3 = 6
7 is a prime!.....

1996 is divisible by 2, Goldbach conjecture: 3 + 1993 = 1996
1997 is a prime!
1998 is divisible by 2, Goldbach conjecture: 5 + 1993 = 1998
1999 is a prime!
2000 is divisible by 2, Goldbach conjecture: 3 + 1997 = 2000
Length of prime list is 303
>>>