Recreational Mathematics · Digit-Power Identities

Narcissistic Digit-Power Representations Generator

Created by Inder J. Taneja. Search for exact identities built from the digits of each number, with or without division and with optional positive and negative terms.

Without division

N = ±a₁^c ± a₂^c ± ··· ± aₖ^c

Here c is the power of every digit. Use the classical choice c = k (the number of digits), or search an exponent range. Example: 153 = 1³ + 5³ + 3³.

With division

N = (±Σ aᵢ^c) / (±Σ aᵢ^d), 0 ≤ d < c

The numerator and denominator use the same digits in their original order. The quotient must be exact; zero denominators are excluded. Example: 37 = (3³ + 7³)/(3 + 7).

Choose the search

Representation type

Without divisionWith division

Signs and exponents

All positivePositive and negative

In the positive and negative mode, each digit term may independently have + or −. For exponent zero, 0⁰ is defined as 1 in this generator.

Without division Use c = number of digits (classical narcissistic numbers)

From NTo NMinimum digit power cMaximum digit power cMaximum results per number and type

What is c? It is the power applied to each digit in the numerator, or to each digit when there is no division. For example, 153 = 1³ + 5³ + 3³ uses c = 3. With division, 37 = (3³ + 7³)/(3¹ + 7¹) uses c = 3 above the fraction line and d = 1 below it. The search tries every c from the minimum to the maximum selected here; for division it tries each d from 0 to c − 1.

The exponent range applies to division and to the nonclassical search without division. A search may cover at most 5,000 numbers and numbers may have at most 7 digits. Broad signed searches can take longer.

Ready.

Verified results

Every identity is checked with integer arithmetic before display and export. Downloads include every result found in the selected interval, including results beyond what fits on screen.