]> Sum of n-th Powers Is an n-th Power

# Sum of n-th Powers Is an n-th Power

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Some positive integers $x 1 x 2 … x n y$ have the following property:

$∑ i = 1 k x i n = x n$

where n is a positive integer. In other words, the sum of n-th powers of k positive integers is equal to the n-th power of a positive integer. This property is relevant in a variety of application domains and has been investigated by many people over time. For an introduction to this topic, see:

This page presents the lowest integers for which the abovementioned property holds for several values of {n,k}. We add another constraint: all positive integers must be different. These values were computed using C and Python programs that I occasionally run to evaluate the performance of various PCs and servers.

Color codes:

• red: the sum comprises only consecutive integers
• green: the number of terms in the sum is lower than the power (i.e., k<n)

## n=2, k=2

These numbers are known as Pythagorean Triples.

  3 2 + 4 2 = 5 2 $5 2 + 12 2 = 13 2$ $8 2 + 15 2 = 17 2$ $7 2 + 24 2 = 25 2$ $20 2 + 21 2 = 29 2$

## n=3, k=3

  3 3 + 4 3 + 5 3 = 6 3 $1 3 + 6 3 + 8 3 = 9 3$ $3 3 + 10 3 + 18 3 = 19 3$ $7 3 + 14 3 + 17 3 = 20 3$ $4 3 + 17 3 + 22 3 = 25 3$ $18 3 + 19 3 + 21 3 = 28 3$ $11 3 + 15 3 + 27 3 = 29 3$ $6 3 + 32 3 + 33 3 = 41 3$

## n=3, k=4

 $1 3 + 5 3 + 7 3 + 12 3 = 13 3$ $5 3 + 7 3 + 9 3 + 10 3 = 13 3$ $2 3 + 3 3 + 8 3 + 13 3 = 14 3$ $4 3 + 7 3 + 8 3 + 17 3 = 18 3$  11 3 + 12 3 + 13 3 + 14 3 = 20 3

## n=4, k=4

 $30 4 + 120 4 + 272 4 + 315 4 = 353 4$ $240 4 + 340 4 + 430 4 + 599 4 = 651 4$

## n=4, k=5

 $4 4 + 6 4 + 8 4 + 9 4 + 14 4 = 15 4$ $4 4 + 21 4 + 22 4 + 26 4 + 28 4 = 35 4$ $1 4 + 8 4 + 24 4 + 36 4 + 38 4 = 45 4$ $1 4 + 2 4 + 12 4 + 24 4 + 44 4 = 45 4$ $4 4 + 6 4 + 31 4 + 44 4 + 46 4 = 55 4$ $2 4 + 13 4 + 16 4 + 44 4 + 48 4 = 55 4$ $2 4 + 14 4 + 28 4 + 33 4 + 52 4 = 55 4$

## n=5, k=4

  27 5 + 84 5 + 110 5 + 133 5 = 144 5

## n=5, k=5

 $19 5 + 43 5 + 46 5 + 47 5 + 67 5 = 72 5$ $21 5 + 23 5 + 37 5 + 79 5 + 84 5 = 94 5$ $7 5 + 43 5 + 57 5 + 80 5 + 100 5 = 107 5$ $78 5 + 120 5 + 191 5 + 259 5 + 347 5 = 365 5$ $79 5 + 202 5 + 258 5 + 261 5 + 395 5 = 415 5$

## n=5, k=6

 $4 5 + 5 5 + 6 5 + 7 5 + 9 5 + 11 5 = 12 5$

## n=5, k=7

 $1 5 + 7 5 + 8 5 + 14 5 + 15 5 + 18 5 + 20 5 = 23 5$

## n=7, k=8

 $12 7 + 35 7 + 53 7 + 58 7 + 64 7 + 83 7 + 85 7 + 90 7 = 102 7$