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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
have the following property:
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.
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n=3, k=3
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n=3, k=4
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n=4, k=4
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n=4, k=5
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n=5, k=4
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n=5, k=5
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n=5, k=6
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n=5, k=7
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n=7, k=8
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