357686312646216567629137 and Friends

Many people close to FWPhys are aware of, in possession of, or about to possess, our first STEM merchandise, the Left Truncatable Prime Pencil. Printed on it is the largest base-10 number that remains prime as you remove digits from the left side by means of sharpening the pencil.

It’s common knowledge and there’s a Wikipedia page dedicated to it. No matter, I wish to post my own code used to generate it. There’s no algorithmic elegance, just a regular tree search.

####################

Base = 10

####################
import numpy as np
import math

import time

# Fast
def Prime(n):
    if n == 0 or n == 1:
        return False
    
    elif n % 2 == 0 and n > 2: 
        return False
    
    return all(n % i for i in range(3, int(math.sqrt(n)) + 1, 2))


NumberLibrary = []

# Initialization
Dig = 0

# Start the tree by finding the prime first digits
for i in range(Base):
    
    if Prime(i):
        NumberLibrary.append(i)

# Iterative tree search
Primality = True

while Primality:
    
    Timestart = time.time()

    Dig += 1
    print(f'Searching and testing digit #{Dig+1}')
    
    Multiplier = Base ** Dig

    NewNumberLibrary = []
    for n in NumberLibrary:
    
        for i in range(1,Base):
            NewNum = n + i * Multiplier
            if Prime(NewNum):
                NewNumberLibrary.append(NewNum)
                
                
    if NewNumberLibrary == []:
        Primality = False;
        print(f'Failed! Time taken: {time.time()-Timestart:.3g} s.')
        print(f"Largest left truncatable prime length is {Dig}")
        
        
    else:
        
        NumberLibrary = NewNumberLibrary
        for k in NumberLibrary:
            if Base != 10:
                print(f"{np.base_repr(k,base = Base)} (base {Base}) = {k} (base 10)")
            if Base == 10:
                print(k)
                
        print(f"{len(NumberLibrary)} candidate(s) found. Time taken: {time.time()-Timestart:.3g} s.")
            
        print("\n")
        
       

And just for fun, here are some lesser left-truncatable primes in other base systems.

Base	Numbers	Value in Base 10
3	212	23
4	321223 233323 333323	3691 3067 4091
5	222232	7817
6	14141511414451435	4836525320399
7	6642623	817337
8	313636165537775	14005650767869
9	2486266212 4284484465 2462868287	988642271 1676456897 977306389
10	357686312646216567629137	–
11	CPU Meltdown
12	CPU Meltdown