Armstrong Number Checker – Digit Powers, Bases and Range Scans
An Armstrong number is a whole number that equals the sum of its own digits, each raised to the power of how many digits the number has. The classic example is 153: it has three digits, and 1³ + 5³ + 3³ = 1 + 125 + 27 = 153. Because the exponent is the digit count, three-digit numbers cube their digits, four-digit numbers raise them to the fourth power, and so on. The same idea travels under several names — narcissistic number, plus perfect digital invariant and pluperfect digital invariant — but the definition never changes. This checker verifies any number instantly and shows the full digit-power calculation so you can see exactly why it passes or fails.
How the Check Works
The tool writes your number out as a digit string, counts the digits to fix the exponent, raises each digit to that power, and adds the results. It then compares the total with the original number: an exact match means the value is an Armstrong number, and any gap is reported as a signed difference so you can tell whether the power sum overshoots or falls short. For 123 the sum is 1³ + 2³ + 3³ = 36, which is 87 short of 123, so 123 is not an Armstrong number. All arithmetic runs on BigInt values rather than floating point, so even the 39-digit champion stays exact instead of drifting once it passes 2^53.
Single-Digit and Multi-Digit Cases
Every single-digit number is trivially an Armstrong number, because a one-digit value has an exponent of 1 and n¹ = n. That is why a scan starting at 1 always lists 1 through 9 before the first interesting hit. Beyond the single digits, the base-10 Armstrong numbers are surprisingly sparse: 153, 370, 371 and 407 are the only three-digit ones, followed by four-digit values such as 1634, 8208 and 9474. The tool handles large multi-digit numbers too — enter 9926315 or 146511208 to watch the seven- and nine-digit cases resolve.
Finding Armstrong Numbers in a Range
Switch to range mode to list every Armstrong number between two bounds, a common classroom and lab exercise. Scanning 1 to 10,000 returns the nine single-digit values plus 153, 370, 371, 407, 1634, 8208 and 9474. Each value in the range is written out, its digits are raised to the digit count and summed, and matches are collected. A guard rail caps a single scan at one million consecutive values so the browser never freezes, and a warning appears before a wide scan on a slower device. Results can be copied, downloaded as text, or exported as CSV with the number, digit count and digit-power sum.
d digits can be at most d × 9^d once every digit is raised to the power d. As d grows, that ceiling eventually rises more slowly than the smallest d-digit number itself, so past 39 digits no Armstrong number can exist. There are exactly 88 of them in base 10.Armstrong Numbers in Other Bases
Being narcissistic is a property of the written form, so it depends on the base. This checker accepts any base from 2 to 36, using the letters A to Z for digit values above 9 exactly as hexadecimal does. The optional base scan lays out bases 2 through 16 side by side for a single value, making it easy to spot numbers that satisfy the digit-power rule in several bases at once or in none beyond the trivial single-digit cases. Every base has its own family of narcissistic numbers, and the general definition — sum of digits raised to the digit count — is identical in each.
Why It Matters
Armstrong numbers are a staple of introductory programming, appearing in coding exercises and technical interviews because they combine digit extraction, exponentiation and a comparison in a few lines of logic. Working through the breakdown here makes the underlying algorithm concrete: split into digits, count them, raise and sum, then compare. Whether you need a quick yes-or-no answer, a full step-by-step explanation for a homework problem, or a complete list across a range, this tool covers each use without a single manual calculation.