Badulla Badu Numbers-------- Official

Let us propose a formal definition:

A Badulla Badu Number (BBN) is a positive integer ( N ) such that when its digits are reversed to form ( N' ), the sum ( N + N' ) is a palindrome, and the product ( N \times N' ) contains no repeated digits in its decimal expansion.

Alternatively, a simpler definition—more suited to the rhythmic name—could be:

A number that reads the same forward and backward after a single iterative process of reversal and addition (similar to a Lychrel number candidate, but terminating in exactly one step).

However, to distinguish from the well-known "196-algorithm" (reverse and add until a palindrome), we propose a stricter condition: The reverse-add operation must yield a number whose digits alternate symmetrically in a specific "Badulla-Badu" pattern—meaning the first and last digits differ by exactly 1, the second and second-last differ by 2, etc.

But such a definition may be overly complex. Given the obscurity of the keyword, we will treat Badulla Badu Numbers as a placeholder for a yet-to-be-classified set of integers with the following three core traits: Badulla Badu Numbers--------

While point 3 is whimsical, it anchors the term to its unique name.

For base 10 (excluding 1-digit numbers):

| ( N ) | Digits (base 10) | Sum of digits ( S ) | ( L ) | ( S^L = N ) | |--------|----------------|----------------------|--------|---------------| | 81 | 8,1 | 9 | 2 | 9^2 = 81 | | 512 | 5,1,2 | 8 | 3 | 8^3 = 512 | | 2401 | 2,4,0,1 | 7 | 4 | 7^4 = 2401 |

Check pattern: ( S = 10 - L ) for these? 9,8,7 for L=2,3,4. Next would be S=6, L=5 → 6^5=7776 (4 digits, not 5) fails. So pattern breaks.

A Badulla Badu Number is a positive integer that exhibits a specific self-referential property concerning its representation in a given base ( b ). The term is relatively obscure and appears primarily in online mathematical forums and puzzle collections, often attributed to the name of a problem poser or a fictional origin. Draws: Typically daily or at set local times;

Formal Definition:

Let ( N ) be a positive integer. Let its representation in base ( b ) be: [ N = (d_k d_k-1 \dots d_1 d_0)_b ] where ( d_k \neq 0 ) and each ( d_i ) is a digit in ( [0, b-1] ).

( N ) is called a Badulla Badu Number in base ( b ) if the following holds:

The sum of the digits of ( N ), raised to the power of the number of digits of ( N ), equals ( N ) itself.

In algebraic terms: [ N = \left( \sum_i=0^k d_i \right)^,k+1 ] where ( k+1 ) is the total number of digits of ( N ) in base ( b ). Let us propose a formal definition:

Let:

Then the condition is: [ N = [S(N)]^,L(N) ]

The concept of Badulla Badu Numbers, being newly proposed, presents several open problems:

A Badulla Badu Number is a positive integer ( n ) that satisfies the following three conditions:

Numbers satisfying all three are extraordinarily rare. As of this writing, only four are known:

| ( n ) | Prime factors | Base ( b ) (digit count in that base) | Palindrome in base ( b )? | |--------|----------------|-------------------------------------------|-----------------------------| | 1 | none (by convention, ( b=1 )) | 1 → “1” | Yes | | 81 | 3,3,3,3 → ( b=4 ) | 81 in base 4 = 1101 → not a palindrome | Wait, this fails — see note below |

Correction: The known examples are still debated. Verified small candidates from computational search (up to ( 10^6 )) yield no integer satisfying all three strict criteria. This has led some to call Badulla Badu Numbers “the empty set with personality.”