Automorphic number¶

A base-b natural number whose numeral reappears as the final k digits of its square, equivalently an idempotent n²≡n modulo b^k for k equal to its digit length.

Version: v1 · 2026-09-28 · History
Domain-specific #: 8087
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Recreational Mathematics, Number Theory → Mathematics

Core Idea¶

An automorphic number in base b is a k-digit natural number n such that n² ends in the same k digits. Algebraically, n²≡n (mod b^k), so n is an idempotent in Z/b^kZ. In decimal, 25 qualifies because 25²=625, and 76 qualifies because 76²=5776.

Factoring x²−x=x(x−1) shows why the prime-power structure of the base matters: for each distinct prime-power factor, an idempotent selects residue 0 or 1, and Chinese-remainder recombination produces branches. Compatible solutions lift through powers of b and correspond to idempotents in the b-adic inverse limit.

The label is representation-dependent. A number can be automorphic in one base and not another; digit length and treatment of 0 or leading zeros must be explicit. Lists should be generated by modular arithmetic or verified squaring rather than pattern continuation.

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Numbers That Hide in Their Squares

Some numbers are special: when you multiply them by themselves, the answer ends with the same number! 25 times 25 is 625, which ends in 25. And 76 times 76 is 5776, which ends in 76. Those are automorphic numbers, at least when we write numbers the usual way.

Numbers Whose Squares End the Same

An automorphic number is a number whose square ends with the number itself. In our usual counting, 5 works because 5 × 5 = 25, 25 works because 25 × 25 = 625, and 76 works because 76 × 76 = 5776. Whether a number is automorphic depends on how you write numbers — a number that works in our base-ten system might not work if you count in a different base. To find these numbers, you should actually check by squaring or by careful math rules, not just guess from a pattern.

Self-Ending Square Numbers

An automorphic number in base b is a k-digit natural number n whose square ends in the same k digits as n. In decimal, 25 (25² = 625) and 76 (76² = 5776) qualify. In modular arithmetic this means n² ≡ n (mod bᵏ), which can be rewritten n(n − 1) ≡ 0. Because n and n − 1 share no common factors, each prime-power piece of bᵏ must divide one or the other, which is why the prime factorization of the base controls how many solutions there are. The property depends on the base, and the digit count and the treatment of 0 or leading zeros must be stated. Lists of them should be generated by modular arithmetic or checked by squaring, not by continuing an apparent digit pattern.

An automorphic number in base b is a k-digit natural number n with n² ending in the same k digits, i.e., n² ≡ n (mod bᵏ), so n is an idempotent of the ring Z/bᵏZ. Decimal examples are 25 (625) and 76 (5776). Factoring n² − n = n(n − 1) explains the role of the base's prime-power structure: for each distinct prime-power factor of bᵏ, an idempotent must be congruent to 0 or 1, and the Chinese remainder theorem recombines these choices into separate branches of solutions. Compatible solutions for successive k lift through powers of b and correspond to idempotents of the b-adic inverse limit. The label is representation-dependent — a number can be automorphic in one base and not another — and digit length and conventions for 0 and leading zeros must be explicit. Enumeration should use modular arithmetic or verified squaring rather than extrapolating digit patterns.

Structural Signature¶

Sig role-phrases:

base. Fixes digits, place values, and modulus powers. Constitutive parameter. If altered: Automorphy can change with the base.
k-digit numeral. Supplies n and the suffix length tested. Constitutive object. If altered: Leading-zero conventions must be stated.
squaring map. Maps n to n². Identity-bearing operation. If altered: Other powers define related but different classes.
suffix congruence. Tests n²−n divisible by b^k. Defining relation. If altered: A visual decimal coincidence without the modulus is insufficient.
compatible lift. Extends residue fixed points across increasing powers toward b-adic idempotents. Structural family relation. If altered: Not every displayed prefix is an independent branch.

What It Is Not¶

Not base-independent. The modulus is a power of the numeral base.
Not any repeated suffix. The suffix must equal the original numeral.
Not a palindrome. Symmetry of digits is irrelevant.
Not only decimal. The definition applies to arbitrary bases.

Scope of Application¶

Automorphic numbers occur in elementary number theory, modular rings, idempotents, Chinese-remainder constructions, p-adic/b-adic analysis, algorithms, and recreational mathematics.

Enumeration. Lists solutions by digit length.
Modular algebra. Studies idempotents modulo base powers.
b-adic limits. Links compatible suffixes to infinite fixed points.
Algorithms. Lifts branches without brute-force squaring.
Recreational mathematics. Explains striking square endings rigorously.

Clarity¶

State base, digit length, leading-zero convention, and whether n is a finite numeral, residue class, or b-adic limit. Verify n²−n divisibility by b^k rather than quoting only the displayed suffix.

Manages Complexity¶

The suffix rule is elementary, while the congruence exposes ring decomposition and inverse limits. This bridge compresses huge square calculations into local prime-power choices and compatible lifting.

Abstract Reasoning¶

Count k digits in the chosen base.
Compute n²−n or its residue modulo b^k.
Factor b to analyze idempotent choices prime-power by prime-power.
Use Chinese remaindering and lifting to extend branches.
Check representation conventions before comparing lists.

Knowledge Transfer¶

The fixed-point method transfers to other polynomial congruences, but automorphic numbers specifically use squaring and their own digit-length modulus. A fixed point of another map is a different sequence.

Examples¶

Canonical¶

In base ten, 76²=5776, so the final two digits equal the two-digit input; equivalently 76²−76 is divisible by 10².

Mapped back: base → 10; k-digit numeral → 76 with k=2; squaring map → 5776; suffix congruence → 5776≡76 mod 100; compatible lift → member of a decimal branch.

Applied / In Practice¶

An enumerator factors b, selects 0/1 idempotents modulo each prime-power component, recombines them with the Chinese remainder theorem, and lifts compatible residues from b^k to b^(k+1).

Mapped back: base → factored b; k-digit numeral → residue at precision k; squaring map → x↦x²; suffix congruence → idempotent equation; compatible lift → next-precision branch.

Structural Tensions¶

T1: digit pattern vs. algebraic identity. The suffix is intuitive while the congruence proves it. Diagnostic: Which modulus exactly matches the digits?

T2: finite numeral vs. b-adic branch. Each number is finite while compatible endings approach an infinite object. Diagnostic: Is the claim about one residue or its limit?

T3: easy verification vs. hard enumeration. Squaring tests one case while listing all cases requires structure. Diagnostic: Have all prime-power branches been counted?

Structural–Framed Character¶

Automorphic number is structural. Bases, congruences, idempotents, and inverse limits are formal; only notation conventions are framed. Its portable skeleton is Fixed Point, related rather than a strict parent because this is a specialized arithmetic class. Evaluative weight is absent; practice dependence is low; origin lies in number theory; vocabulary travels with an explicit base. Its character: a numeral-level fixed point of squaring under base-power truncation.

Structural Core vs. Domain Accent¶

Skeletal core. An object remains unchanged after a transformation viewed through a finite-resolution quotient.

Domain-bound accent. Numeral bases, suffixes, squares, modular rings, and b-adic lifts define automorphy.

Why not prime. Fixed points travel, but automorphic numbers are one arithmetic realization.

Fixed Point. Squaring leaves the residue unchanged.
Modularity. Equality is evaluated modulo a base power.
No strict DAG edge is added.

Neighborhood in Abstraction Space¶

Automorphic number sits in a moderately populated region (47^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Abstract Algebra & Category Theory (24 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Trimorphic number. Tell: Is the fixed power two or three?
Cyclic number. Tell: Is multiplication rotation or square suffix involved?
Idempotent residue. Tell: Is a residue class or digit-length numeral meant?
Palindrome. Tell: Does the square preserve suffix or mirror digits?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Automorphic_number (revision 1287067229).
Preserved source candidate: http://www.numericana.com/answer/p-adic.htm#decimal

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.