Automorphic Number Checker
Check whether a number is automorphic, meaning its square ends in the number itself, such as 5, 6, 25 and 76.
Last reviewed by the Radiatus Cloud team
Check whether a number is automorphic (its square ends in itself).
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Check for an automorphic number
An automorphic number is a number whose square ends in the same digits as the number itself. This checker squares the number and tests whether the square finishes with the original digits. For instance, twenty-five is automorphic because its square, six hundred and twenty-five, ends in twenty-five, and seventy-six is automorphic because its square, five thousand seven hundred and seventy-six, ends in seventy-six. These curious numbers are also called circular or self-reproducing numbers.
The single-digit automorphic numbers are one, five and six.
Self-reproducing squares
Automorphic numbers have a surprising property: there are automorphic numbers of every length, and they can be extended digit by digit, with the five-family and six-family continuing indefinitely into ever longer automorphic numbers. This links them to a concept in advanced number theory involving so-called ten-adic numbers. For most people, though, they are simply a delightful arithmetic curiosity to discover and verify.
Very large numbers can exceed the precision of ordinary arithmetic, so this checker is best used with moderate values. All calculation happens locally in your browser.
About this tool
Because the automorphic number checker runs entirely in your browser, you can use it as many times as you like for free, with nothing uploaded and no sign-up required. Everything is computed locally on your own device, so it is fast and private.
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Frequently Asked Questions
What is an automorphic number?
A number whose square ends in the same digits as the number, such as twenty-five, whose square six hundred and twenty-five ends in twenty-five.
What are the smallest automorphic numbers?
The single-digit ones are one, five and six, followed by twenty-five and seventy-six among the two-digit numbers.
Why are they called self-reproducing?
Because the number reappears at the end of its own square, as if it reproduces itself, which also gives them the name circular numbers.
Is there a limit to this checker?
Very large numbers exceed ordinary arithmetic precision, so it is best used with moderate values where the square is computed exactly.
Privacy & Security
Everything runs in your browser; nothing is uploaded.
How to Use
Enter a positive integer to check if it is automorphic.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.