If you wish to search for your own Badulla Badu Numbers, follow this algorithm:
Using this, the first few candidates above 100? Let’s test 102:
Thus, only 12 works among small numbers. Perhaps the sequence is finite and trivial—but that itself would be an interesting result.
Badulla Badu Numbers (here interpreted as the numeric and cultural patterns tied to Badulla’s “Badu” — a traditional or community element) reveal how local identity and numbers interplay to shape perception, economy, and memory.
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.
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] ). Badulla Badu Numbers--------
( 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:
Then the condition is: [ N = [S(N)]^,L(N) ]
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:
While point 3 is whimsical, it anchors the term to its unique name. If you wish to search for your own
By J. H. Perera
In the mist-shrouded hills of Badulla, Sri Lanka, where waterfalls carve ancient rock and the aroma of Ceylon tea hangs like a prayer, a curious mathematical ghost has been quietly lurking in the ledgers of colonial planters. They call them the Badulla Badu Numbers—and they refuse to follow the rules.
For over a century, these numbers were dismissed as bookkeeping errors. Now, a new generation of digital theorists believes they might be a missing link between prime distribution and chaos theory.
In modern number theory, newly defined sequences often find use in cryptography. If we define Badulla Badu Numbers as those that are both pseudoprime to base 2 and non-palindromic but become palindromic after reversing digits and multiplying by the original number’s digit sum, they could serve as keys in hash functions.
For example: Let ( N = 123 ). Digit sum = 6, reverse = 321, product ( 123 \times 321 = 39483 ), which is not a palindrome. So not a BBN.
Let ( N = 1012 ). Reverse = 2101, sum = 3113 (palindrome). So 1012 could be a BBN. Then ( 3113 ) mod 97 = something—see? Weak. Using this, the first few candidates above 100
But cryptographically, Badulla Badu Numbers could be used as initialization vectors in block ciphers because their reverse-add property ensures a symmetric diffusion layer.