Automorphic Number Calculator: Finally Understand Numbers That Live in Their Own Square

Let me tell you about the first time I saw an automorphic number. I was reading about mathematical curiosities, and I learned that 5² = 25 (ends with 5), 6² = 36 (ends with 6), 25² = 625 (ends with 25), and 76² = 5776 (ends with 76).

I thought, "That's fascinating—these numbers are like mathematical ouroboros, self-contained in their own squares."

These are called automorphic numbers (or circular numbers). They appear to be rare, but once you know the pattern, you can generate infinitely many of them. In fact, there are infinite automorphic numbers in base 10!

In this guide, I'll walk you through everything you need to know about automorphic numbers—from the simple 5 and 6 to the famous 376 and 625, and even the 10,000-digit behemoths.

Ready to explore self-squared numbers? Try our Automorphic Number Calculator and discover which numbers live inside their own squares.

What Is an Automorphic Number?

An automorphic number (also called a circular number) is a number whose square ends with the number itself.

The Formula

For a number n with k digits:

n² ≡ n (mod 10ᵏ)

Or more simply: n² ends with the digits of n

Simple Examples

Why Are They Called "Automorphic"?

The term comes from Greek:

• autos (αὐτός) = self
• morphē (μορφή) = form, shape

So "automorphic" means "self-shaped" — the number's shape appears within its own square.

Alternative Names

All Known Automorphic Numbers

One-Digit Automorphic Numbers

Two-Digit Automorphic Numbers

Three-Digit Automorphic Numbers

Four-Digit Automorphic Numbers

Actually let me list properly:

The Pattern

Automorphic numbers come in pairs that sum to 10ᵏ + 1:

• 5 + 6 = 11 (10¹ + 1)
• 25 + 76 = 101 (10² + 1)
• 376 + 625 = 1001 (10³ + 1)
• 9376 + 0625 = 10001 (10⁴ + 1)
• 109376 + 890625 = 1,000,001 (10⁶ + 1)

Infinite Automorphic Numbers

There are infinitely many automorphic numbers in base 10.

How to Generate Them

For any k-digit automorphic number n, there is a (2k)-digit automorphic number ending with n.

Example: Starting from 5 (1-digit):

• 5 → 25 (2-digit)
• 25 → 625 (3-digit? Actually 625 is 3-digit, but 25 is 2-digit)
• 625 → 0625? Wait, need careful.

Actually the sequence:

• 5 (1-digit)
• 25 (2-digit) ends with 5
• 625 (3-digit) ends with 25
• 0625? But 0625 is just 625.

Better example: 376 (3-digit) → 9376 (4-digit) → 109376 (6-digit) → 7109376 (7-digit)...

The Two Infinite Series

There are two infinite series of automorphic numbers:

Series A (ending in 5):

• 5
• 25
• 625
• 0625 (but leading zero dropped, so 625 again? Actually 625 is 3-digit, next is 90625?)

Let me correct: The two series are:

1. ...376 (ending in 376)
2. ...625 (ending in 625)

For 5-digit: 90625 and 09376 → 90625, 9376 (but 9376 is 4-digit)

Actually known infinite series:

• ...376 (like 376, 9376, 109376, 7109376, ...)
• ...625 (like 625, 0625, 90625, 890625, 2890625, ...)

Step-by-Step Examples

Example 1: Check if 5 is Automorphic

Step 1: Identify the number

• n = 5

Step 2: Calculate the square

• 5² = 25

Step 3: Check if square ends with the number

• 25 ends with "5"? ✓

Result: 5 is AUTOMORPHIC! ✨

Example 2: Check if 6 is Automorphic

Step 1: n = 6

Step 2: 6² = 36

Step 3: 36 ends with "6"? ✓

Result: 6 is AUTOMORPHIC! ✨

Example 3: Check if 25 is Automorphic

Step 1: n = 25

Step 2: 25² = 625

Step 3: 625 ends with "25"? ✓

Result: 25 is AUTOMORPHIC! ✨

Example 4: Check if 76 is Automorphic

Step 1: n = 76

Step 2: 76² = 5,776

Step 3: 5,776 ends with "76"? ✓

Result: 76 is AUTOMORPHIC! ✨

Example 5: Check if 376 is Automorphic

Step 1: n = 376

Step 2: 376² = 141,376

Step 3: 141,376 ends with "376"? ✓

Result: 376 is AUTOMORPHIC! ✨

Example 6: Check if 7 is Automorphic

Step 1: n = 7

Step 2: 7² = 49

Step 3: 49 ends with "7"? ✗ (ends with 49)

Result: 7 is NOT automorphic

Visual Pattern: 376 and 625

These two 3-digit automorphic numbers are complementary:

Notice: 376 + 625 = 1,001 (10³ + 1)

The Magic of 376

376 × 376 = 141,376 (ends with 376) 376 × 376 × 376 = 53,157,376 (ends with 376!)

In fact, 376ⁿ always ends with 376 for any positive integer n.

The Magic of 625

625² = 390,625 (ends with 625) 625³ = 244,140,625 (ends with 625!)

Like 376, 625ⁿ always ends with 625.

Properties of Automorphic Numbers

Property 1: n and n² Share Last k Digits

For a k-digit automorphic number n, n² mod 10ᵏ = n.

Property 2: Complement Pairs

If n is a k-digit automorphic number, then (10ᵏ + 1 - n) is also automorphic.

Examples:

• 5 (1-digit) → 10¹ + 1 - 5 = 6
• 25 (2-digit) → 10² + 1 - 25 = 76
• 376 (3-digit) → 10³ + 1 - 376 = 625
• 9376 (4-digit) → 10⁴ + 1 - 9376 = 625? Actually 10,001 - 9,376 = 625, but 625 is 3-digit. The 4-digit version is 0625.

Property 3: Infinite Length

There are automorphic numbers of any length (by adding digits to the left).

Property 4: Only End in 0, 1, 5, or 6

Any automorphic number greater than 1 must end in 5 or 6.

Why? For a number to be automorphic, n² mod 10 = n mod 10. The only digits satisfying x² ≡ x (mod 10) are 0, 1, 5, 6.

Check:

• 0² = 0 ✓
• 1² = 1 ✓
• 2² = 4 ≠ 2 ✗
• 3² = 9 ≠ 3 ✗
• 4² = 16 ≠ 4 ✗
• 5² = 25 → ends with 5 ✓
• 6² = 36 → ends with 6 ✓
• 7² = 49 → ends with 9 ≠ 7 ✗
• 8² = 64 → ends with 4 ≠ 8 ✗
• 9² = 81 → ends with 1 ≠ 9 ✗

Property 5: Powers Preserve Automorphism

If n is automorphic, then nᵏ also ends with n for all positive integers k.

Example: 376² ends with 376, 376³ ends with 376, etc.

How to Use Our Automorphic Calculator

Step 1: Enter a Number

Type any non-negative integer. Examples: 5, 25, 76, 376, 625

Step 2: Click Check Number

The calculator:

• Calculates the square (using BigInt for large numbers)
• Checks if the square ends with the original number
• Displays the result with visual highlighting

Step 3: Read Your Results

You'll see:

• Verdict: Automorphic or not
• Square calculation: n² displayed
• Visual highlighting: The matching ending digits are highlighted
• Explanation: Plain language description

Example Buttons

Quick test known automorphic numbers:

• 0, 1, 5, 6 (1-digit)
• 25, 76 (2-digit)
• 376, 625 (3-digit)

What It Handles

The Mathematics Behind Automorphic Numbers

Solving n² ≡ n (mod 10ᵏ)

This congruence can be rewritten as:

n(n - 1) ≡ 0 (mod 10ᵏ)

Since n and n-1 are coprime, we need:

• n ≡ 0 (mod 2ᵏ) and n ≡ 1 (mod 5ᵏ)
• OR n ≡ 1 (mod 2ᵏ) and n ≡ 0 (mod 5ᵏ)

These two solutions give the two infinite series of automorphic numbers:

• The series ending in ...376
• The series ending in ...625

Example: k=3 (1000)

Solution 1: n ≡ 0 (mod 8) and n ≡ 1 (mod 125) → n = 376 Solution 2: n ≡ 1 (mod 8) and n ≡ 0 (mod 125) → n = 625

Known Automorphic Numbers (Extended)

The 10,000-Digit Automorphic Number

There exist automorphic numbers with thousands of digits. The 10,000-digit automorphic number has been computed and verified.

Fun Facts About Automorphic Numbers

The 5 and 6 Connection

5 and 6 are the only single-digit automorphic numbers (besides 0 and 1). They sum to 11 (10¹ + 1).

376 is Magic

376 appears in many mathematical curiosities. 376² = 141,376, 376³ = 53,157,376, and so on—all end with 376.

625 is Magic Too

625² = 390,625, 625³ = 244,140,625, all end with 625.

The Russian Doll Pattern

Each automorphic number is contained within the next larger one:

• 5 ⊂ 25 ⊂ 625 ⊂ 0625 ⊂ 90625...
• 6 ⊂ 76 ⊂ 376 ⊂ 9376 ⊂ 109376...

Used in Puzzles

Automorphic numbers appear in math puzzles and recreational mathematics because of their surprising property.

Common Mistakes

Mistake 1: Forgetting 0 and 1

Wrong: "5 and 6 are the only automorphic numbers under 10" Right: 0 and 1 are also automorphic (0²=0, 1²=1).

Mistake 2: Thinking All Numbers Ending in 5 Are Automorphic

Wrong: "15 ends with 5, so it must be automorphic" Right: 15² = 225, ends with 25, not 15. Only specific numbers ending in 5 work.

Mistake 3: Thinking All Numbers Ending in 6 Are Automorphic

Wrong: "16 ends with 6, so it must be automorphic" Right: 16² = 256, ends with 56, not 16.

Mistake 4: Confusing with Armstrong Numbers

Wrong: "153 is automorphic because it's an Armstrong number" Right: Automorphic means square ends with the number, not sum of digit powers.

Mistake 5: Negative Numbers

Wrong: "-5 is automorphic" Right: Automorphic numbers are defined for non-negative integers.

Quick Reference

Automorphic Condition

n² ≡ n (mod 10ᵏ) where k = number of digits of n

First 10 Automorphic Numbers

0, 1, 5, 6, 25, 76, 376, 625, 9376, 90625

Last Digit Rule

Automorphic numbers > 1 must end in 5 or 6.

Complement Pairs

Powers Property

If n is automorphic, nᵏ ends with n for all positive integers k.

Frequently Asked Questions

What is an automorphic number?

A number whose square ends with the number itself. Example: 25² = 625 ends with 25.

Why are they called automorphic?

From Greek: "autos" (self) + "morphe" (form) = self-shaped.

What are the first few automorphic numbers?

0, 1, 5, 6, 25, 76, 376, 625, 9376, 90625...

Is 0 automorphic?

Yes, 0² = 0 ends with 0.

Is 1 automorphic?

Yes, 1² = 1 ends with 1.

Is 2 automorphic?

No, 2² = 4 ends with 4, not 2.

Is 5 automorphic?

Yes, 5² = 25 ends with 5.

Is 6 automorphic?

Yes, 6² = 36 ends with 6.

Are there infinitely many automorphic numbers?

Yes, there are infinitely many in base 10.

How does your calculator handle large numbers?

It uses BigInt for precise integer arithmetic, handling numbers of any size.

Your Turn: Start Exploring

Automorphic numbers are a fascinating mathematical curiosity—numbers that literally appear inside their own squares.

Here's your practice plan:

1. Start with the basics: 0, 1, 5, 6 (all automorphic)
2. Try two-digit: 25, 76 (both work!)
3. Test three-digit: 376, 625 (both work!)
4. Check non-automorphic: 2, 3, 4, 7, 8, 9, 10, 12
5. Use example buttons: 0, 1, 5, 6, 25, 76, 376, 625
6. Watch the pattern: Notice all automorphic numbers > 1 end in 5 or 6
7. Try larger numbers: 9376, 90625, 109376

Ready to start? Open up our Automorphic Number Calculator and try it yourself. Start with 5, then 25, then 376.

You'll discover the magic of self-squared numbers faster than you think.

Have questions? Stuck on a particular number? Drop a comment below or reach out. I've been where you are, and I'm happy to help.

— The Solvezi Team

Disclaimer: This calculator is for educational purposes. For very large numbers (>15 digits), calculations may take a moment.