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.

Version: v1 · 2026-09-28 · History
Domain-specific #: 11252
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Recreational Number Theory → Mathematics

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 $n=\sum_{i=0}^{k-1}d_i b^i$, define $F_b(n)=\sum_i d_i^{d_i}$ under an explicit convention for $0^0$. Then $n$ is perfect when $F_b(n)=n$. 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.

The property depends on numeral representation. The same integer has different digits in different bases and can therefore be fixed in one base but not another. The $0^0=1$ and $0^0=0$ conventions also change cases containing zero and the treatment of zero itself, so a list is incomplete without both parameters. The number 1 is a trivial fixed point in every base. Periodic points of $F_b$ form sociable digit-to-digit invariants; a two-cycle is amicable, while a perfect invariant is the period-one case.

Only finitely many fixed points occur for any fixed base. A $k$-digit input grows at least like $b^{k-1}$, whereas $F_b(n)$ is at most $k(b-1)^{b-1}$; beyond a base-dependent length the exponential growth of place value dominates the linear number of bounded digit terms, forcing the map downward until it enters a finite state set and eventually cycles. The abstraction is a base- and convention-dependent fixed point of a digit-power dynamical system, not ordinary numerical perfection or invariance under changing notation.

Structural Signature¶

Sig role-phrases:

the numeral base — integer base $b$ fixing the available digits and place values
the digit expansion — representation of a natural number as ordered digits $d_i$
the zero-power convention — explicit choice for $0^0$ governing cases containing zero
the self-power transform — each digit mapped to $d_i^{d_i}$
the digit-sum map — transformed contributions added without retaining place position
the fixed-point equality — output of the map exactly equal to the original integer
the representation dependence — status changing when the same integer is written in another base
the finiteness bound — place-value growth eventually exceeding the maximum linear sum of bounded digit powers
the dynamical extension — periodic orbits classified as sociable, with period one defining perfection
the concept boundary — separation from ordinary perfect numbers and notation-invariant numerical properties

What It Is Not¶

Not a perfect number in divisor theory. Perfection here means a fixed point of a digit-power map, not equality with the sum of proper divisors.
Not invariant under changing base. The same integer acquires different digits and can lose or gain the property in another numeral system.
Not defined without a zero convention. Treating zero to the zero as zero or one changes cases containing zero.
Not any digit-power equality. Each digit must be raised to its own value and the sum must reproduce the original represented number.
Not a nontrivial cycle of longer period. Amicable and sociable digit invariants return after multiple iterations; a perfect invariant returns after one.
Not evidence of infinitely many solutions at fixed base. Place-value growth eventually dominates the bounded per-digit contribution, leaving only a finite search region.
Not numeral-independent mathematical essence. The defining dynamics act on representation, making base and convention essential parameters.

Scope of Application¶

Perfect digit-to-digit invariant applies to fixed points of the base-b digit map that sums d raised to d under an explicit convention for zero raised to zero.

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.
Zero conventions. Candidates containing zero depend on whether 0^0 is defined as one or another value.
Related invariant families. Amicable and sociable digit-to-digit cycles extend the same map beyond fixed points.
Applicability boundary. These are not perfect numbers or narcissistic numbers, and a search without a growth bound is not a proof that no larger solutions exist; “Munchausen number” is mnemonic rather than the definition.

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. The sharper number-theoretic question is which bounds on digit count make exhaustive classification possible for the stated base and convention.

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. This compression replaces open-ended enumeration of natural numbers with base-specific bounds and exact evaluation, while making representation dependence explicit so results from one base or zero convention are never imported silently into another.

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. A number satisfying another digit-power identity or entering the fixed point later is not itself a perfect digit-to-digit invariant.

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. Its boundary is representation dependence: the property is not intrinsic to the integer across bases, “perfect” is classificatory rather than evaluative, and numerical coincidences do not create a dynamical or algebraic mechanism outside the defining digit map.

Examples¶

Canonical¶

In base ten, take 3435 and transform each digit by raising it to itself: 3³+4⁴+3³+5⁵=27+256+27+3125=3435. The result equals the original integer, so 3435 is a perfect digit-to-digit invariant under the decimal self-power map. Place position disappears during summation even though it determined the original numeral. A number containing zero cannot be tested unambiguously until the chosen convention for 0⁰ is stated. Rewriting the same integer in another base can also change its status.

Mapped back: Decimal is the numeral base, 3435 the digit expansion, exponentiation the self-power transform, and addition the digit-sum map. The equality is the fixed-point equality; alternative bases expose the representation dependence, and zeros require the zero-power convention.

Applied / In Practice¶

A program enumerates all invariants in base b. It first proves a digit-length bound: the smallest k-digit numeral eventually exceeds k(b−1)^(b−1), so only finitely many lengths require search. It then iterates the transform for each candidate and separately records fixed points and longer cycles. Tests pin the 0⁰ convention and verify results by reconstructing digits in the selected base. The program never mixes these objects with classical perfect numbers defined by divisor sums.

Mapped back: The growth comparison supplies the finiteness bound. Iteration implements the digit-sum map over the digit expansion; fixed points and cycles separate the fixed-point equality from the dynamical extension. Base conversion preserves the representation dependence, and divisor-sum exclusion enforces the concept boundary.

Structural Tensions¶

T1 — Identity versus admissible variation. Perfect digit-to-digit invariant must remain recognizable across legitimate variants. Admissible variation is bounded by this condition: Exact fixed points provide striking examples of representation-dependent arithmetic dynamics. The stable element is expressed by this 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. Treating every surface change as a new abstraction fragments the identity, while allowing a change to the constitutive relation produces a false positive.

Diagnostic: After the proposed variation, can an analyst still establish this 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?

T2 — Recognition versus proxy. The domain needs observable or inferential evidence for Perfect digit-to-digit invariant, but the evidence is not automatically the identity. The working recognition rule is: the concept boundary — separation from ordinary perfect numbers and notation-invariant numerical properties. A familiar indicator can occur without the defining relation, and the relation can persist when a customary detector is unavailable.

Diagnostic: Does the evidence establish the defining claim—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—or only a correlated sign?

T3 — Definition versus operational judgment. A compact definition aids reuse, whereas actual classification in recreational number theory can require expert decisions about boundary conditions, measurements, conventions, or exceptions. The property depends on numeral representation. The definition must constrain those judgments without pretending that every admissible case can be recognized from a label alone.

Diagnostic: Which observation would make a competent practitioner reject the classification under the stated definition?

T4 — Scope versus overextension. Perfect digit-to-digit invariant has a genuine habitat in which exact fixed points provide striking examples of representation-dependent arithmetic dynamics. Yet These are not perfect numbers or narcissistic numbers, and a search without a growth bound is not a proof that no larger solutions exist; “Munchausen number” is mnemonic rather than the definition. A useful application map therefore has to be broad enough to cover recurring practice and narrow enough to exclude merely topical or metaphorical occurrences.

Diagnostic: Can the claimed application fill the same carrier and relation roles, or has only the name traveled?

T5 — Transfer versus domain accent. Knowledge about Perfect digit-to-digit invariant can travel within its home domain, and some structural lessons may travel farther. 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. What transfers must be separated from the specialist vocabulary, warrant, and closure conditions that remain anchored in recreational number theory.

Diagnostic: Is the receiving case a literal instance of Perfect digit-to-digit invariant, a co-instance of Classification, or only an analogy?

T6 — Autonomy versus reduction. Perfect digit-to-digit invariant is a strict specialization of Classification, but the edge does not erase the domain differentia. The broader node supplies only the necessary structural relation; recreational number theory supplies the carrier, warrant, boundary, and exception conditions expressed by this identity: 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. The entry is over-split if those conditions add no discriminating work and under-specified if the parent alone is used for cases that require them.

Diagnostic: Can a domain expert use the added conditions to distinguish Perfect digit-to-digit invariant from another case that equally instantiates Classification?

Structural–Framed Character¶

Perfect digit-to-digit invariant is mixed: structurally specifiable but materially dependent on its disciplinary frame. Its structural side consists of the carrier the numeral base — integer base $b$ fixing the available digits and place values and the constitutive relation 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. Its framed side comes from recreational number theory, which fixes what the terms denote, what counts as evidence, and when a qualification or exception defeats the classification.

Across the principal tests, the entry is not merely a free-floating pattern. Evaluative weight: the identity can be stated descriptively even when its use has practical or normative consequences. Practice dependence: the concept boundary — separation from ordinary perfect numbers and notation-invariant numerical properties. Institutional stabilization: disciplinary conventions may stabilize the name and test without necessarily creating every underlying event or relation. Vocabulary portability: the invariant is 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. Import versus recognition: an outside case qualifies literally only if the same typed roles and collapse condition are available; otherwise the comparison is analogical.

The reusable remainder is Classification under a reviewed subsumption relation. That node preserves the necessary cross-domain organization after the recreational number theory-specific carrier, evidence, and exceptions are removed. Perfect digit-to-digit invariant remains autonomous because its recognition and collapse conditions distinguish cases that the parent alone leaves together.

Structural Core vs. Domain Accent¶

What is skeletal. The portable skeleton is a typed carrier organized by a constitutive relation, an invariant, a recognition test, and a collapse condition. Here the carrier is the numeral base — integer base $b$ fixing the available digits and place values. The decisive relation is 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, which also states the controlling invariant at this level. Stripped of specialist nouns, this organization is represented by Classification.

What is domain-bound. recreational number theory supplies the actual objects or agents, admissible transformations, units or conventions, standards of warrant, and named exceptions. In this case, recognition requires evidence for the concept boundary — separation from ordinary perfect numbers and notation-invariant numerical properties. Admissible variation is bounded by the condition that exact fixed points provide striking examples of representation-dependent arithmetic dynamics, and the classification collapses when perfection here means a fixed point of a digit-power map, not equality with the sum of proper divisors. These are constitutive differentia, not illustrative decoration.

Why it remains a domain-specific node. The reviewed DAG relation is subsumption to Classification. Outside recreational number theory, the parent captures only the reusable structural remainder. The specialist name remains literal only where the concept boundary — separation from ordinary perfect numbers and notation-invariant numerical properties can be established under the domain's standards of warrant.

This entry is a kind of Classification.

Immediate parent — Classification (subsumption). 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. The parent supplies the necessary broader identity—Sorting entities into discrete categories by explicit rules, turning unbounded variation into a finite, reusable map for downstream reasoning and action.—while the candidate adds the source-domain carrier, recognition rule, and failure conditions. The defining source account begins: 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.
Nearest catalog surface declined — Perfect number. Its rematch score was 0.183011. Retrieval proximity did not establish synonymy or parentage; the carrier, invariant, and collapse condition remain different.
Related reasoning operations. Evidence, comparison, boundary testing, and representation can support a case without becoming additional DAG parents.

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 (47^th 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

Not to Be Confused With¶

Classification. This is the reviewed immediate parent or structural prerequisite, not a synonym. Tell: retain Perfect digit-to-digit invariant only when the domain-specific relation 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. and its source-domain warrant are established; otherwise route the case to Classification.
Descartes Number. This is the closest catalog retrieval surface, not an accepted synonym or parent. Tell: Ask which entry's carrier, invariant, and collapse test the case actually satisfies; shared vocabulary or a score of 0.731174 is insufficient.
Not a perfect number in divisor theory. Perfection here means a fixed point of a digit-power map, not equality with the sum of proper divisors. Tell: Require the positive recognition condition that the concept boundary — separation from ordinary perfect numbers and notation-invariant numerical properties.
Not invariant under changing base. The same integer acquires different digits and can lose or gain the property in another numeral system. Tell: Replace the familiar surface feature and test whether 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.
A detector, representation, or consequence. A method may reveal Perfect digit-to-digit invariant, a notation may describe it, and an outcome may follow from it without any of those being identical to the abstraction. Tell: Would the defining relation remain if the present detector, notation, or downstream effect changed?
A metaphorical transfer. A case outside the home domain may resemble the structure while lacking its native role types and standards of warrant. Tell: If only the general organization survives, route the comparison to Classification rather than treating it as another Perfect digit-to-digit invariant instance.

References¶

Frozen Wikipedia revision: https://en.wikipedia.org/wiki/Perfect_digit-to-digit_invariant (revision 1322694177).
Supporting reference preserved in the packet: https://books.google.com/books?id=h30pAgAAQBAJ&pg=PA136
Supporting reference preserved in the packet: https://arxiv.org/abs/0911.3038
Supporting reference preserved in the packet: https://books.google.com/books?id=AOu2AwAAQBAJ&pg=PT28
Supporting reference preserved in the packet: http://www.magic-squares.net/narciss.htm
Supporting reference preserved in the packet: http://www.numberphile.com/videos/3435.html
Supporting reference preserved in the packet: https://web.archive.org/web/20170413072736/http://www.numberphile.com/videos/3435.html

The frozen Wikipedia revision is discovery provenance. The cited source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; URL transport failure alone was not treated as substantive contradiction.