I'm working on problem 46 from project euler:

It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.

9 = 7 + 2×12    
15 = 7 + 2×22
21 = 3 + 2×32
25 = 7 + 2×32
27 = 19 + 2×22
33 = 31 + 2×12

It turns out that the conjecture was false. What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?

My code is so ridiculously inefficient and it takes minutes before an answer pops up. (Boundary numbers are just rough estimates)

numbers = [(2*x**2) for x in list(range(1, 1000))]
normal = list(range(1, 10000000))

#produces primes under n
def primes(n):
    sieve = [True] * n
    for i in range(3, int(n**0.5) + 1, 2):
        if sieve[i]:
            sieve[i * i:: 2 * i] = [False] * int((n - i * i - 1)/(2 * i) + 1)
    return [2] + [i for i in range(3, n, 2) if sieve[i]]

primes = primes(1000)
final = []

#add the two lists in every way possible
added = [x + y for x in numbers for y in primes]

#if it does not appear in a normal number line send it to final 
for x in added:
    if x not in normal:
        final.append(x)

print(min(final))

I also lack the knowledge to use any mathematical tricks or algorithms. Where can I start learning code efficiency/performance and simple algorithms to use in my code?

I'm working on problem 46 from project euler:

It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.

9 = 7 + 2×1^2    
15 = 7 + 2×2^2
21 = 3 + 2×3^2
25 = 7 + 2×3^2
27 = 19 + 2×2^2
33 = 31 + 2×1^2

It turns out that the conjecture was false. What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?

My code is so ridiculously inefficient and it takes minutes before an answer pops up. (Boundary numbers are just rough estimates)

numbers = [(2*x**2) for x in list(range(1, 1000))]
normal = list(range(1, 10000000))

#produces primes under n
def primes(n):
    sieve = [True] * n
    for i in range(3, int(n**0.5) + 1, 2):
        if sieve[i]:
            sieve[i * i:: 2 * i] = [False] * int((n - i * i - 1)/(2 * i) + 1)
    return [2] + [i for i in range(3, n, 2) if sieve[i]]

primes = primes(1000)
final = []

#add the two lists in every way possible
added = [x + y for x in numbers for y in primes]

#if it does not appear in a normal number line send it to final 
for x in added:
    if x not in normal:
        final.append(x)

print(min(final))

I also lack the knowledge to use any mathematical tricks or algorithms. Where can I start learning code efficiency/performance and simple algorithms to use in my code?

I'm working on problem 46 from project euler:

It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square. 9 = 7 + 2×12

15 = 7 + 2×22

21 = 3 + 2×32

25 = 7 + 2×32

27 = 19 + 2×22

33 = 31 + 2×12

It turns out that the conjecture was false. What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?

My code is so ridiculously inefficient and it takes minutes before an answer pops up. (Boundary numbers are just rough estimates)

numbers = [(2*x**2) for x in list(range(1, 1000))]
normal = list(range(1, 10000000))

#produces primes under n
def primes(n):
    sieve = [True] * n
    for i in range(3, int(n**0.5) + 1, 2):
        if sieve[i]:
            sieve[i * i:: 2 * i] = [False] * int((n - i * i - 1)/(2 * i) + 1)
    return [2] + [i for i in range(3, n, 2) if sieve[i]]

primes = primes(1000)
final = []

#add the two lists in every way possible
added = [x + y for x in numbers for y in primes]

#if it does not appear in a normal number line send it to final 
for x in added:
    if x not in normal:
        final.append(x)

print(min(final))

I also lack the knowledge to use any mathematical tricks or algorithms. Where can I start learning code efficiency/performance and simple algorithms to use in my code?

I'm working on problem 46 from project euler:

It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.

9 = 7 + 2×12    
15 = 7 + 2×22
21 = 3 + 2×32
25 = 7 + 2×32
27 = 19 + 2×22
33 = 31 + 2×12

It turns out that the conjecture was false. What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?

My code is so ridiculously inefficient and it takes minutes before an answer pops up. (Boundary numbers are just rough estimates)

numbers = [(2*x**2) for x in list(range(1, 1000))]
normal = list(range(1, 10000000))

#produces primes under n
def primes(n):
    sieve = [True] * n
    for i in range(3, int(n**0.5) + 1, 2):
        if sieve[i]:
            sieve[i * i:: 2 * i] = [False] * int((n - i * i - 1)/(2 * i) + 1)
    return [2] + [i for i in range(3, n, 2) if sieve[i]]

primes = primes(1000)
final = []

#add the two lists in every way possible
added = [x + y for x in numbers for y in primes]

#if it does not appear in a normal number line send it to final 
for x in added:
    if x not in normal:
        final.append(x)

print(min(final))

I also lack the knowledge to use any mathematical tricks or algorithms. Where can I start learning code efficiency/performance and simple algorithms to use in my code?