Suppose that, for . denotes the sum of the digits of n. Thus all of yield . That is, has “generators”. Numbers with more than one generator (i.e. ) are called “Junction Numbers” by Kaprekar.
Let and be two “Junction Numbers”, such that they have equal number of digits, say, digits. If be the generators of and be the generators of .
give the generators of .
Let us consider an example to understand this theorem.
Consider two Junction Numbers 519 (generated by both 498 and 507) and 521 (generated by both 499 and 508).
Number of digits in or or .
Then according to the theorem,
give the four generators of .
is a number with 115 digits
Kaprekar named this theorem as Kaprekar’s Last Theorem in 1962, when he was seriously ill and feared that his death was nearing. He miraculously recovered and named it as Junction Combination Theorem.
Smallest “Junction Numbers”
- Smallest “Junction Number” with two generators is
- Smallest “Junction Number” with three generators is or
- Smallest “Junction Number” with four generators is or