Project Euler 10: Summation of primes in Go
Algorithm : really important

What you want to to is to use the Sieve of Eratosthenes because you already know an upper bound for the biggest number you're going to consider. Getting all the primes smaller than 2 millions is going to be straightforward.  This will solve your process took too long issue. 

Other things not as important


You could probably write the following function :

func CheckIfIsNewPrime(number uint64) bool {
    check_if_number_is_new_prime := false
    var remainder uint64 = 0

    for i := range primes {
        remainder = number % primes[i]
        if remainder == 0 {
            check_if_number_is_new_prime = false
            break
        } else {
            check_if_number_is_new_prime = true
        }
    }
    return check_if_number_is_new_prime
}


in a more concise way (no real impact on performance)

func CheckIfIsNewPrime(number uint64) bool {
    for i := range primes {
        if number % primes[i] == 0 {
            return false
        }
    }
    return true
}

I don't know anything about Go, but I guess

if CheckIfIsNewPrime(counter) == true {


could be written as

if CheckIfIsNewPrime(counter) {



As you go further in Project Euler, you'll see that most of the problem is to find the algorithm you want to use and not really to implement it.