## Many numbers can be expressed as the sum of a square and a cube. Some of them in more than one way.

Consider the palindromic numbers that can be expressed as the sum of a square and a cube, both greater than $1$, in exactly $4$ different ways.

For example, $5229225$ is a palindromic number and it can be expressed in exactly $4$ different ways:

$2285^2 + 20^3$

$2223^2 + 66^3$

$1810^2 + 125^3$

$1197^2 + 156^3$

Find the sum of the five smallest such palindromic numbers.

### To find the palindromic numbers that can be expressed as the sum of a square and a cube in exactly 4 different ways, we can follow the steps outlined below:

1. Initialize an empty list to store the palindromic numbers.

2. Starting with a loop, iterate through all numbers greater than 1 (to exclude the trivial cases).

3. For each number, calculate the number of ways it can be expressed as the sum of a square and a cube.

4. If the number of ways is exactly 4, check if the number is palindromic.

5. If it is palindromic, add it to the list of palindromic numbers.

6. Once we have found the five smallest palindromic numbers, calculate the sum and return it.

Here is the Python code to solve the problem:

“`python

def is_palindrome(n):

# Function to check if a number is palindromic

return str(n) == str(n)[::-1]

def find_palindromic_numbers():

# Function to find the five smallest palindromic numbers

palindromic_numbers = [] # List to store the palindromic numbers

count = 0 # Number of palindromic numbers found

number = 2 # Start with 2 to exclude trivial cases

while count < 5: # Keep searching until we find five numbers total_ways = 0 # Number of ways the number can be expressed as sum of a square and a cube # Iterate through all possible pairs of square and cube for square in range(2, int(number**0.5) + 1): for cube in range(2, int(number**(1/3)) + 1): if square**2 + cube**3 == number: total_ways += 1 if total_ways == 4: # If the number has four ways, check if it is palindromic if is_palindrome(number): palindromic_numbers.append(number) count += 1 # Increment the count of palindromic numbers found number += 1 # Move to next number return sum(palindromic_numbers) # Call the function to find the sum of the five smallest palindromic numbers sum_palindromic_numbers = find_palindromic_numbers() print(sum_palindromic_numbers) ``` When you run this code, it will output the sum of the five smallest palindromic numbers that can be expressed as the sum of a square and a cube in exactly 4 different ways.

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