]> 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 $(x_{1} x_{2} \dots x_{n} y)$ have the following property:

\sum_{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}