Automorphic number: n² ends with n. 1-digit: 1,5,6. 2-digit: 25,76. 3-digit: 376,625. In base 10, exactly two d-digit automorphic numbers exist for each d≥1 (their last digits sum to 10^d+1). Connected to p-adic numbers and Hensel's lemma.
How to Use
Enter n:
- Auto?: Does n² end in n?
- Square: Full n² value
- Trimorphic?: Does n³ end in n?
Patterns
For d-digit automorphic numbers: exactly 2 per digit count. They come in pairs summing to 10^d+1: 5+6=11, 25+76=101, 376+625=1001. Extend via Hensel lifting.
Mathematical Theory
n is automorphic iff n²≡n (mod 10^d) where d=digits(n). Equivalently: n(n−1)≡0 (mod 10^d). Since 10^d=2^d·5^d, this factors into conditions mod 2^d and mod 5^d via CRT.
Step-by-Step Instructions
- 1Enter number.
- 2Check automorphic.
- 3View square.
- 4Check trimorphic.
- 5Explore sequence.