The Miracle

Number Theory Level 3

Those who believe in miracles are as great as miracles themselves. -- From an unknown book

There are many dates when there are miracles. Suppose the date is expressed as an 8-digit number, 1st~4th represents the year, 5th~6th represents the month, 7th~8th represents the day, when the corresponding digit isn't enough, use 0 to pad right, and such date exists in real world. For instance, 20191002 is a valid date , while 20190229 is not. The year ranges from 0001 to 9999.

After observation, the dates when miracles take place have these properties (ignore the zeros before the digits):

  • The number composed by digits represing the day is a prime number . For instance, for 20191002, 02 is a prime number.

  • The number composed by digits represing the month and day is a prime number . For instance, for 20191002, 1002 is not a prime number.

  • The whole date is a prime number . For instance, for 20191002, 20191002 is not a prime number.

Now, you need to figure out how many valid dates are there in total when miracles take place.

Note: Be careful to check whether it is a leap year or not. Year 3200, 6400, 9600 are considered as leap years.


The answer is 55157.

Alice Smith by Alice Smith , 20, China

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1 solution

David Vreken
Oct 3, 2019

The following Python code outputs an answer of 55157 \boxed{55157} . 

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# The Miracle
# Python 2.7.3

import datetime
from datetime import timedelta

# function to test if a number is prime
def isprime(n):
    n = abs(int(n))
    if n < 2:
        return False
    if n == 2: 
        return True    
    if not n & 1: 
        return False
    for x in range(3, int(n ** 0.5) + 1, 2):
        if n % x == 0:
            return False
    return True

# function to test if a date is a valid date
def vdate(strdate):
    # get year, month, and date from strdate
    y = strdate[0:4]
    m = strdate[4:6]
    d = strdate[6:8]
    # test if the date is a valid date using the isprime function
    v = False
    if isprime(int(d)):
        if isprime(int(m + d)):
            if isprime(int(strdate)):
                v = True
    # return the results
    return v

# go through each date from 00010101 to 99991231 and count how many
#   how many times it's valid
vtotal = 0
date = datetime.datetime(1, 1, 1)
while True:
    strdate = date.isoformat().replace("-", "")[0:8]
    if vdate(strdate):
        # if it is a valid date then add one to the running total
        vtotal += 1
    if strdate == "99991231":
        # if it is the last date then stop the loop
        break
    # get the next date
    date += timedelta(1)

# display the results
print "Total Valid Dates:", vtotal

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