Full Prime

The naive approach will be looping through all number smaller than $10^{100}$ and check if they are full primes.

We can do better. By definition, every suffix must be a prime. Hence, from the base primes $2,3,5,7$ , we can recursively generate a list of full primes by adding a digit in front of the full primes we found. This can be done with either Depth-First Search or Breadth-First Search. My approach is using a kind of BFS with the the number of digits it possess as depth.

Surprisingly, there are no full primes with 25 digits.

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from random import randint

def pow(a,d,n):
    ans = 1
    while d != 0:
        if d & 1:
            ans = (ans * a) % n
        d >>= 1
        a = (a * a) % n
    return ans

def isPrime(n, k):
    if n < 2: return False
    if n < 4: return True
    if n % 2 == 0: return False

    s = 0
    d = n-1
    while d & 1 == 0:
        s += 1
        d >>= 1

    for i in range(k):
        a = randint(2, n-2)    # 2 <= a <= n-2
        x = pow(a,d,n)
        if x == 1: continue
        for j in range(s):
            if x == n-1: break
            x = (x * x) % n
        else:
            return False
    return True

full_prime = [2,3,5,7]

ans = 0
while full_prime:    # full_prime is not empty -> continue generating full primes
    ans += sum(full_prime)

    # generate full primes by adding another a digit in front of the full primes found
    full_prime = [int(str(j)+str(i)) for i in full_prime for j in range(1,10)
                  if isPrime(int(str(j)+str(i)),100)]

print ans

This code runs in about 14 seconds.