In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most nn-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.
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.[1]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, 7 = 4 + 1 + 1 + 1.[1]Gauss proved the triangular case in 1796, commemorating the occasion by writing in his diary the line "ΕΥΡΗΚΑ!num = Δ + Δ + Δ",[2] and published a proof in his book Disquisitiones Arithmeticae. For this reason, Gauss's result is sometimes known as the Eureka theorem.[3] The full polygonal number theorem was not resolved until it was finally proven by Cauchy in 1813.[1] The proof of Nathanson (1987) is based on the following lemma due to Cauchy:
For odd positive integers a and b such that b2 < 4a and 3a < b2 + 2b + 4 we can find nonnegative integers s, t, u, and v such that a = s2 + t2 + u2 + v2 and b = s + t + u + v.
^Ono, Ken; Robins, Sinai; Wahl, Patrick T. (1995), "On the representation of integers as sums of triangular numbers", Aequationes Mathematicae, 50 (1–2): 73–94, doi:10.1007/BF01831114, MR 1336863, S2CID 122203472.
Heath, Sir Thomas Little (1910), Diophantus of Alexandria; a study in the history of Greek algebra, Cambridge University Press, p. 188.
Nathanson, Melvyn B. (1987), "A short proof of Cauchy's polygonal number theorem", Proceedings of the American Mathematical Society, 99 (1): 22–24, doi:10.2307/2046263, JSTOR 2046263, MR 0866422.
Nathanson, Melvyn B. (1996), Additive Number Theory The Classical Bases, Berlin: Springer, ISBN978-0-387-94656-6. Has proofs of Lagrange's theorem and the polygonal number theorem.
August 23, 2023
fermat, polygonal, number, theorem, confused, with, fermat, last, theorem, additive, number, theory, states, that, every, positive, integer, most, gonal, numbers, that, every, positive, integer, written, three, fewer, triangular, numbers, four, fewer, square, . Not to be confused with Fermat s Last Theorem 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 Contents 1 Examples 2 History 3 See also 4 Notes 5 ReferencesExamples EditThree 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 Edit 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 1 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 7 4 1 1 1 1 Gauss proved the triangular case in 1796 commemorating the occasion by writing in his diary the line EYRHKA num D D D 2 and published a proof in his book Disquisitiones Arithmeticae For this reason Gauss s result is sometimes known as the Eureka theorem 3 The full polygonal number theorem was not resolved until it was finally proven by Cauchy in 1813 1 The proof of Nathanson 1987 is based on the following lemma due to Cauchy For odd positive integers a and b such that b2 lt 4a and 3a lt b2 2b 4 we can find nonnegative integers s t u and v such that a s2 t2 u2 v2 and b s t u v See also EditPollock s conjectures Waring s problemNotes Edit a b c Heath 1910 Bell Eric Temple 1956 Gauss the Prince of Mathematicians in Newman James R ed The World of Mathematics vol I Simon amp Schuster pp 295 339 Dover reprint 2000 ISBN 0 486 41150 8 Ono Ken Robins Sinai Wahl Patrick T 1995 On the representation of integers as sums of triangular numbers Aequationes Mathematicae 50 1 2 73 94 doi 10 1007 BF01831114 MR 1336863 S2CID 122203472 References EditWeisstein Eric W Fermat s Polygonal Number Theorem MathWorld Heath Sir Thomas Little 1910 Diophantus of Alexandria a study in the history of Greek algebra Cambridge University Press p 188 Nathanson Melvyn B 1987 A short proof of Cauchy s polygonal number theorem Proceedings of the American Mathematical Society 99 1 22 24 doi 10 2307 2046263 JSTOR 2046263 MR 0866422 Nathanson Melvyn B 1996 Additive Number Theory The Classical Bases Berlin Springer ISBN 978 0 387 94656 6 Has proofs of Lagrange s theorem and the polygonal number theorem Retrieved from https en wikipedia org w index php title Fermat polygonal number theorem amp oldid 1150374055, wikipedia, wiki, book, books, library,
