Die ersten 10 Automorphe Zahlen

This family came up through recreational arithmetic rather than research. The one-digit cases 0, 1, 5 and 6 were called circular numbers for a long time before anyone extended the idea, and the older literature sometimes used circular and spherical for the related property of a number whose every power ends in the same digit. Wikipedia still records "circular number" as an alternative name. The term automorphic — formed on itself — came into general mathematical use around the early 1940s, and is usually traced to Maurice Kraitchik's Mathematical Recreations of 1942, which did a great deal to popularise it. The provenance before that point is thin, and claims that particular ancient civilisations knew the pattern should be treated with caution.

What rescues the topic from triviality is the algebra underneath. Asking for a k-digit number whose square ends in itself is asking for a solution of x² ≡ x modulo 10^k — an idempotent in the ring of integers modulo 10^k. Since 10 factors as 2 · 5, the Chinese remainder theorem splits that ring in two, and the number of idempotents is exactly 2 raised to the number of distinct prime factors of the base: four in base 10, namely 0, 1, and a complementary pair.

Push k upwards and the digits of each pair member stop changing: 6, 76, 376, 9376, 109376, 7109376 are successive truncations of a single infinite object. That object is an idempotent of the ring of 10-adic integers, the number system Kurt Hensel (1861–1941) arrived at in 1897 while adapting Weierstrass's power-series methods to algebraic functions and developed systematically in his Theorie der algebraischen Zahlen of 1908. The schoolroom curiosity and Hensel's lemma turn out to be the same statement seen from two ends.

Wichtige Eigenschaften

• A number n is automorphic exactly when n² ≡ n modulo 10^k, where k is the digit count of n.
• In base 10 there are precisely four solutions of x² ≡ x (mod 10^k) for every k ≥ 1 — in general 2^ω(b) solutions in base b, where ω counts the distinct prime factors of b.
• The two non-trivial solutions modulo 10^k always sum to 10^k + 1: 376 + 625 = 1001, and 9376 + 625 = 10001.
• Every automorphic number other than 0 and 1 ends in 5 or in 6, and the two families are exactly those two complementary solutions.
• Some lengths have only one automorphic number, because the other solution carries a leading zero: the length-4 pair is 9376 and 0625, so only 9376 counts.
• The sequence begins 0, 1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376 and continues indefinitely, gaining about one digit per term.
• Each term agrees with the next in all its digits, so the two families converge digit-by-digit on the two non-trivial idempotents of the 10-adic integers.