The Second Smallest Three-Digit Palindromic Square Number

In the realm of number theory, palindromic square numbers hold a special place. A palindromic square number is both a square of an integer and reads the same forwards and backwards. This article explores the second smallest three-digit palindromic square number that begins with the digit 4 and has a digit sum of 16.

Understanding Palindromic Squares

A three-digit palindromic square number can be represented in the form aba, where a and b are digits. Not all three-digit numbers are palindromic squares, and identifying them requires both mathematical analysis and a bit of exploration.

Identifying Relevant Squares

Let's identify the three-digit palindromic squares. We need to find integers such that their squares read the same forwards and backwards. Here are the relevant squares:

121 11^2 484 22^2 676 26^2 729 27^2 - not a palindrome 841 29^2 - not a palindrome 900 30^2 - not a palindrome

From this list, we can see that the only relevant palindromic squares are 121 and 484.

Filtering the Precise Candidate

We are specifically interested in the second smallest three-digit palindromic square number that starts with the digit 4. Given the list above, the only number that fits this criterion is 484.

Verifying the Sum of Digits

To ensure that 484 meets the additional condition, we calculate the sum of its digits:

4 8 4 16

Since 484 satisfies both the palindromic square condition and the digit sum condition, we can confidently state that the number is 484.

Exploring Beyond the Solution

For a deeper understanding, let's explore how we can find the smallest such numbers. Starting with a leading digit, we need to check squares of numbers that begin with 1, 2, and 3, and then move to 4. Here’s why we rule out leading digits 2 and 3:

For a number to start with 2, its square must end with 4. There are no such numbers. For a number to start with 3, its square must end with 9. There are no such numbers.

Similarly, for a number to start with 4, it must end with 2 or 8 because these are the only digits that square to produce a number ending in 4. Hence, 22^2 484 is the valid candidate as it adheres to both the palindromic square and digit sum conditions.

No smaller solution exists because the 200’s and 300’s are devoid of palindrome squares, and no other two-digit squares meet the digit sum requirement when squared.

Interestingly, the next smallest solution can be derived by multiplying the smallest solution by 2. Since 22 is the smallest number ending in 2 or 8, its square 484 is the next palindromic square meeting the criteria.

This exploration not only identifies the desired number but also provides a deeper insight into the properties of palindromic square numbers and their digit characteristics.