# Goldbach's Conjecture

I am doing an exercise on finding the nearest pair of prime numbers p₁ and p₂ such that:

• p₁ ≤ p₂
• p₁ + p₂ = n (for 4 ≤ n ≤ 10⁷)
• p₁ and p₂ are primes
• p₂ - p₁ is minimal

I think it should work perfectly except that it got a TLE. Can someone point out how can I avoid it?

import math
import random

def _try_composite(a, d, n, s):
if pow(a, d, n) == 1:
return False
for i in range(s):
if pow(a, 2 ** i * d, n) == n - 1:
return False
return True  # n  is definitely composite

def is_prime(n, _precision_for_huge_n=16):
if n in _known_primes:
return True
if n in (0, 1) or any((n % p) == 0 for p in _known_primes):
return False
d, s = n - 1, 0
while not d % 2:
d, s = d >> 1, s + 1
# Returns exact according to http://primes.utm.edu/prove/prove2_3.html
if n < 1373653:
return not any(_try_composite(a, d, n, s) for a in (2, 3))
if n < 25326001:
return not any(_try_composite(a, d, n, s) for a in (2, 3, 5))

_known_primes = [2, 3]
_known_primes += [x for x in range(5, 1000, 2) if is_prime(x)]

def main():
n = int(input())
if n > 4:
for smaller in range(n // 2, -1, -1):
if n - smaller >= smaller:
if is_prime(n - smaller) and is_prime(smaller):
print (smaller, n - smaller)
flag = True
break
else:
print ('2 2')

main()

• Since I need to minimize p_2 - p_1, it should be from n/2 to 3 and thus doesn't change? – Gareth Ma Mar 13 '18 at 2:29
• Also using a sieve is a good idea... except sieving from 1 to 10^7 takes around 19.3 second of my code. – Gareth Ma Mar 13 '18 at 2:29
• – Gareth Ma Mar 13 '18 at 9:57
• @TobySpeight Sorry, I did not copy it. I typed it out, so it is a typo. – Gareth Ma Mar 15 '18 at 4:43
• Typing it yourself is the form of copying that I was thinking of - thanks for correcting the description. It's good to know it's not a bug in the code! – Toby Speight Mar 15 '18 at 8:42

My solution consists of a single change:

• Changed the is_prime function into a prime sieve (from this answer) + lookup

My final code (Also handled odd numbers just for fun, definitely not because I misread the question):

def main():
import itertools
import math
izip = itertools.zip_longest
chain = itertools.chain.from_iterable
compress = itertools.compress
def rwh_primes2_python3(n):
""" Input n>=6, Returns a list of primes, 2 <= p < n """
zero = bytearray([False])
size = n//3 + (n % 6 == 2)
sieve = bytearray([True]) * size
sieve = False
for i in range(int(n**0.5)//3+1):
if sieve[i]:
k=3*i+1|1
start = (k*k+4*k-2*k*(i&1))//3
sieve[(k*k)//3::2*k]=zero*((size - (k*k)//3 - 1) // (2 * k) + 1)
sieve[  start ::2*k]=zero*((size -   start  - 1) // (2 * k) + 1)
ans = [2,3]
poss = chain(izip(*[range(i, n, 6) for i in (1,5)]))
ans.extend(compress(poss, sieve))
return ans

string2 = "Impossible"
n = int(input())
sieve = [t for t in rwh_primes2_python3(n) if t <= math.floor((n // 2) / 2) * 2 + 1][::-1]
another = [t for t in rwh_primes2_python3(n) if t >= math.floor((n // 2) / 2) * 2 + 1]

if n > 5 and n % 2 == 0:
for smaller in sieve:
if n - smaller in another:
print (smaller, n - smaller)
break
elif n % 2 == 1 and n != 5:
if n - 2 in another or n-2 in sieve: print (2, n-2)
else: print ('Impossible')
elif n == 4:
print ('2 2')
else:
print ('2 3')

main()

• One advantage of this approach is that if you want to test multiple n, you just sieve for the largest one, and that cost is amortized over all the inputs. I should point out that you invoke rwh_primes2_python3(n) twice - I think you can call it just once, and then split the result into upper and lower halves (actually just one half, for the for smaller in another loop - assuming primes get sparser, that's a better choice - and we can keep sieve complete for the if n-smaller in sieve test). – Toby Speight Mar 14 '18 at 8:39