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Fermat polygonal number theorem

Every positive integer is a sum of at most n n-gonal numbers

Fermat polygonal number theorem
Summary

Every positive integer is a sum of at most n n-gonal numbers

In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most n n-gonal numbers. That is, every positive integer can be written as the sum of three or fewer triangular numbers, and as the sum of four or fewer square numbers, and as the sum of five or fewer pentagonal numbers, and so on. That is, the n-gonal numbers form an additive basis of order n.

Examples

Three such representations of the number 17, for example, are shown below:

  • 17 = 10 + 6 + 1 (triangular numbers)
  • 17 = 16 + 1 (square numbers)
  • 17 = 12 + 5 (pentagonal numbers).

History

[[Gauss's diary]] entry related to sum of triangular numbers (1796)

The theorem is named after Pierre de Fermat, who stated it, in 1638, without proof, promising to write it in a separate work that never appeared. Joseph Louis Lagrange proved the square case in 1770, which states that every positive number can be represented as a sum of four squares, for example, . Gauss proved the triangular case in 1796, commemorating the occasion by writing in his diary the line "ΕΥΡΗΚΑ! ", and published a proof in his book Disquisitiones Arithmeticae. For this reason, Gauss's result is sometimes known as the Eureka theorem.{{citation

For odd positive integers a and b such that b2 2 + 2b + 4 we can find nonnegative integers s, t, u, and v such that and .

Notes

References

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References

  1. {{harvtxt. Heath. 1910.
  2. Bell, Eric Temple. (1956}}. Dover reprint, 2000, {{ISBN). "The World of Mathematics". [[Simon & Schuster]].
Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

additive-number-theory analytic-number-theory figurate-numbers theorems-in-number-theory
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