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 n whose square ends in n, equivalently n²≡n modulo b^k. It is a base-dependent modular idempotent and can lie on a compatible b-adic branch. 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. 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.

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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.

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. Use it with base, digit length, leading-zero convention, exact congruence, finite-residue versus b-adic status, and distinction from palindromic, cyclic, or other-power digit phenomena explicit.

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. The closest near miss sets the boundary: Idempotents modulo b^k are the exact algebraic class; the numeral label adds base and digit convention.

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. The central digit pattern–algebraic identity tradeoff is this: The suffix is intuitive while the congruence proves it. A second finite numeral–b-adic branch tension matters because Each number is finite while compatible endings approach an infinite object. The easy verification–hard enumeration tension adds that Squaring tests one case while listing all cases requires structure.

Abstract Reasoning¶

Use three linked moves: 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. As a collapse test, the case exits when the suffix length is wrong, the base changes, or leading zeros are smuggled in inconsistently. A fourth check is to use Chinese remaindering and lifting to extend branches. A final check is to 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. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. Squaring leaves the residue unchanged. Equality is evaluated modulo a base power.

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