Perfect digit-to-digit invariant¶
A perfect digit-to-digit invariant is a positive integer equal to the sum obtained by raising each of its decimal digits to the power of that same digit.
Core Idea¶
A perfect digit-to-digit invariant in base \(b\) is a natural number that is fixed by the map sending each digit to that digit raised to its own power and summing the results. If \(n=\sum{i=0}^{k-1}di b^i\), define \(Fb(n)=\sumi di^{di}\) under an explicit convention for \(0^0\). Then \(n\) is perfect when \(Fb(n)=n\). In decimal, 3435 is an example because \(3^3+4^4+3^3+5^5=3435\). The popular name “Munchausen number” alludes to each digit metaphorically raising itself.
Scope of Application¶
-
Recreational number theory. Exact fixed points provide striking examples of representation-dependent arithmetic dynamics.
-
Digit-map iteration. Orbits, cycles, basins, and period-one points are studied under one declared base and map.
-
Finite search proofs. Exponential place value eventually exceeds the linear maximum digit contribution, yielding a completeness bound.
-
Exhaustive computation. Algorithms enumerate candidates within the proved bound and verify every equality exactly.
-
Base dependence. Changing numeral representation changes digit decomposition and can change invariant status.
Clarity¶
Perfect digit-to-digit invariant denotes a fixed point of a representation-dependent digit map: sum each base-\(b\) digit raised to itself and recover the original integer. The base and convention for \(0^0\) are part of the definition, so a list without them is incomplete. The term distinguishes perfect fixed points from numbers that enter a cycle only after iteration and from other digit-power invariants with a common exponent.
Manages Complexity¶
Perfect digit-to-digit invariance converts a large integer search into a fixed-point problem bounded by digit count. The analyst tracks base, zero-power convention, number of digits, and the maximum possible sum of digit self-powers. Once positional growth exceeds that maximum, no larger candidate length can work, leaving a finite search. Fixed points and longer cycles form separate branches under iteration.
Abstract Reasoning¶
Bounding move. Compare the largest possible digit-self-power sum for k base-b digits with the smallest k-digit number to infer a finite maximum candidate length. Evaluation move. Apply the map exactly and retain numbers equal to their image as fixed points. Convention move. Separate results under zero-to-zero equals zero versus one and under each numeral base. Dynamics move. Iterate nonfixed values to distinguish transients and longer cycles from perfect invariants. Boundary move.
Knowledge Transfer¶
Within the home domain. Perfect digit-to-digit invariants transfer across recreational number theory and digit-dynamics research when a base-specific digit function is summed and fixed points satisfy equality with the original integer. Base, digit multiset, exponent or mapping, fixed-point equation, and search bounds retain exact roles. Beyond the home domain (C — formal object). The construction applies literally to any radix and digit function that are explicitly defined.
Relationships to Other Abstractions¶
Current abstraction Perfect digit-to-digit invariant Domain-specific
Parents (1) — more general patterns this builds on
-
Perfect digit-to-digit invariant is a kind of Classification Prime
Perfect digit-to-digit invariant is a domain-specific kind of Classification: A perfect digit-to-digit invariant is a positive integer equal to the sum obtained by raising each of its decimal digits to the power of that same digit.
Hierarchy path (1) — routes to 1 parentless root
- Perfect digit-to-digit invariant → Classification
Neighborhood in Abstraction Space¶
Perfect digit-to-digit invariant sits in a moderately populated region (47th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Numeral Systems & Digit Algorithms (11 abstractions)
Nearest neighbors
- Sum-product number — 0.94
- Multiplicative Digital Root — 0.87
- Cyclic Number — 0.87
- Average Order of an Arithmetic Function — 0.85
- Signedness — 0.85
Computed from structural-signature embeddings · 2026-10-08