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Topic: Fermat polygonal number theorem

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Triangular number

Polygonal number

Square (geometry)

1770

1771

	What's Special About This Number?
		is the number of planar partitions of 10.
		is the number of planar partitions of 11.
		is the number of planar partitions of 12.
	www.stetson.edu /~efriedma/numbers.html (7324 words)

	PlanetMath: polygonal number
		Polygonal numbers were studied somewhat by the ancients, as far back as the Pythagoreans, but nowadays their interest is mostly historical, in connection with this famous result:
		In other words, any nonnegative integer is a sum of three triangular numbers, four squares, five pentagonal numbers, and so on.
		This is version 2 of polygonal number, born on 2003-09-02, modified 2003-09-03.
	planetmath.org /encyclopedia/PolygonalNumber.html (215 words)

	Pierre de Fermat Summary
		Fermat provided his own proof of the three-line theorem, and in doing so he made use of analytical methods, determining points in a plane by two coordinates, showing that if the coordinates are related by equations like 2x + 3y = 5, then the point lies on a straight line, and so on.
		Fermat developed numerous theorems involving prime numbers and integral numbers and, consequently, is regarded as the founder of modern number theory.
		Fermat, who added the aristocratic "de" to his name in his early 1630s, was the son of Dominique Fermat, a successful leather merchant, and Claire de Long, who came from a highly respected family of lawyers.
	www.bookrags.com /Pierre_de_Fermat (5076 words)

	[No title]
		Number theory is the study of the divisibility properties of the integers.
		Fermat showed that a prime number of the form 4k+1 occurs as the hypotenuse of a right triangle, but a prime of the form 4k+3 is never the hypotenuse of a right triangle.
		Fermat conjectured that every positive integer is a sum of at most three triangular numbers; every natural number is a sum of at most four square numbers; every natural number is a sum of at most five pentagonal numbers; and so on for hexagonal, heptagonal, and other polygonal numbers.
	www.math.columbia.edu /~rama/chapters/chap1.html (1200 words)

	Reference.com/Encyclopedia/Lagrange's four-square theorem
		Adrien-Marie Legendre improved on the theorem in 1798 by stating that a positive integer can be expressed as the sum of three squares iff it is not of the form 4
		This number is eight times the sum of the divisors of n if n is odd and 24 times the sum of the odd divisors of n if n is even.
		Lagrange's four-square theorem is a special case of the Fermat polygonal number theorem and Waring's problem.
	www.reference.com /browse/wiki/Lagrange's_four-square_theorem (257 words)

	Springer Online Reference Works
		Fermat in about 1630 in the margins of his copy of the book Aritmetika [1] by Diophantus as follows: "It is impossible to partition a cube into two cubes, or a biquadrate into two biquadrates, and in general any power greater than the second into two powers with the same exponent" .
		is infinite or finite (by Jensen's theorem the number of irregular prime numbers is infinite [4]).
		Frey and J.-P. Serre, showed that Fermat's last theorem is implied by the Weil–Taniyama conjecture in the theory of elliptic curves (cf.
	eom.springer.de /F/f038390.htm (1202 words)

	Abramovich, Fujii, and Wilson article
		Therefore the number of dots in the parallelogram is n(n+1), and the number of dots in the triangle is
		, are triangular; numbers 5151, 501501, 50015001, 5000150001,...
		An example of this is Bachet's theorem actually presented in a GSP sketch of Figure 10 (the hexagon serves as a model of a polygonal number of side m).
	jwilson.coe.uga.edu /Texts.Folder/AFW/AFWarticle.html (8744 words)

	Pythagoras and the Pythagoreans
		The number 2 was not originally regarded as a prime number, or even as a number at all.
		In addition, the number a was classified as abundant or deficient according as their divisors summed greater or less than a, respectively.
		Fermat (1630) conjectured that all numbers of this kind are prime.
	www.math.tamu.edu /~don.allen/history/pythag/pythag.html (2531 words)

	Dodecagonal number - Wikipedia, the free encyclopedia
		A dodecagonal number is a figurate number that represents a dodecagon.
		Dodecagonal numbers consistently alternate parity, and in base 10, their units place digits follow the pattern 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.
		By the Fermat polygonal number theorem, every number is the sum of at most 12 dodecagonal numbers.
	en.wikipedia.org /wiki/Dodecagonal_number (172 words)

	History of Mathematics--Spring 2005/HWK5
		Fermat wanted only integer solutions; nowadays when we talk of a diophantine equation we mean one of which we require integer solutions.
		Fermat got in touch with Mersenne through a friend of his who moved to Paris and who got to meet Mersenne and the members of Mersenne's circle.
		Also in 1660 Fermat heard that the great Dutch physicist and mathematician Huygens was in Paris and he wrote to Huygens that he would like to meet him, and he would assuredly go to Paris if his health allowed it; which may have been a hint to Huygens that Huyens should travel to Toulouse.
	www.math.fau.edu /schonbek/HistMath/histmathsp05h8.html (1657 words)

	Al-Farisi biography
		Rashed discusses the claims of Boyer and others that the innovation in the theory of the rainbow was from al-Shirazi, but gives sound arguments for his claim that ascribing the theory to al-Shirazi is unconvincing.
		In fact al-Farisi's approach is based on the unique factorisation of an integer into powers of prime numbers, and, according to Rashed, he states and attempts to prove this, the so-called fundamental theorem of arithmetic, in this work.
		The pair of amicable number 17296, 18416 are known as Euler's amicable pair.
	www-gap.dcs.st-and.ac.uk /~history/Biographies/Al-Farisi.html (1169 words)

	[No title] (Site not responding. Last check: 2007-09-08)
		Fermat certainly claimed it as a theorem, but this is familiar territory:
		The triangular numbers case is easily seen to be equivalent to the 8k+3 case of the theorem, due to Gauss, that a positive integer N is the sum of three squares if and only if it is not of the form 4^a (8k+7).
		He uses but does not prove Dirichlet's theorem on primes in arithmetic progressions, so you might also want something that provides a proof of that, e.g.
	www.math.niu.edu /~rusin/known-math/00_incoming/polygonal (305 words)

	Polygonal Numbers
		Many facts and theorems are known about polygonal numbers, especially of the squares and triangulars.
		In 1638, Fermat proposed that every positive integer is a sum of at most three triangular numbers, four square numbers, five pentagonal numbers, and n n-polygonal numbers.
		Fermat said that all whole numbers can be expressed as the sum of four, or fewer, square numbers.
	www.trottermath.net /numthry/polynos.html (1364 words)

	Augustin Louis Cauchy Summary
		Of special merit in the more than 500 papers that appeared after 1838 were treatises on the mechanics of continuous media, the first rigorous proof of Taylor's theorem, a remarkably modern representation of complex numbers in terms of polynomial congruences, and a collection of papers on the theory of substitutions.
		If the worth of a mathematician were to be measured by the number of times his name appeared in modern college textbooks, Cauchy might be ranked as the greatest of them all.
		He was the first to prove the Fermat polygonal number theorem.
	www.bookrags.com /Augustin_Louis_Cauchy (5064 words)

	Triangular Number (Site not responding. Last check: 2007-09-08)
		Pentagonal Number is 1/3 of a triangular number.
		Fibonacci Numbers which are triangular are 1, 3, 21, and 55 (Ming 1989), and the only
		Pell Number which is triangular is 1 (McDaniel 1996).
	www.math.sdu.edu.cn /mathency/math/t/t306.htm (304 words)

	MathLinks Math Forums :: View topic - Fermat
		Does anyone have a proof of Fermat's polygonal number theorem..that all positive integers are expressible as the sum of at most n n-gonal numbers, first proved by Cauchy.
		I think a proof for Fermat polygonal number theorem can be found from Nathanson, M. "A Short Proof of Cauchy's Polygonal Number Theorem." Proc.
		Well, I've read Nathonson and must admit that the prove of the whole theorem is a bit longer (and most time nasty computation...).
	www.mathlinks.ro /Forum/viewtopic.php?t=15408 (632 words)

	Amazon.ca: Elementary Number Theory in Nine Chapters: Books: James J. Tattersall (Site not responding. Last check: 2007-09-08)
		Historical approach: the book begins with the earliest number theory, that is, polygonal numbers and prime numbers; it has a lot of historical references and anecdotes, and gives some credits to the contributions of China, Iran, etc.
		There are a considerable number of minor typographical errors, but nothing you can't correct yourself.
		Overall, this is a very good number theory textbook for classroom use or self-study.
	www.amazon.ca /Elementary-Number-Theory-Nine-Chapters/dp/0521615240 (695 words)

	[No title]
		You may fly to poetry and to music, and quantity and number will face you in your rhythms and your octaves.
		primes: Fundamental Theorem of Arithmetic, Sieve of Eratosthenes, infinitude of primes, Goldbach Conjecture;
		congruence, linear congruences and the Chinese Remainder Theorem;
	newton.uor.edu /FacultyFolder/Beery/M245CourseInfo.html (986 words)

	Amazon.com: "pentagonal numbers": Key Phrase page (Site not responding. Last check: 2007-09-08)
		M3818, the pentagonal numbers: n 1 2 3 4 5 6 7 8 an 1 5 12 22 35 51 70 92...
		Elementary Number Theory with Applications by Thomas Koshy
		every number is a pentagonal number or the sum of two, three, four, or five pentagonal numbers; and so on for hexagonal numbers, heptagonal numbers,...
	www.amazon.com /phrase/pentagonal-numbers (566 words)

	Table of contents for Library of Congress control number 2001042958
		1 The Theorem of Pythagoras 1 1.1 Arithmetic and Geometry..........
		35 3 Greek Number Theory 37 3.1 The Role of Number Theory.....
		64 5 Number Theory in Asia 66 5.1 The Euclidean Algorithm.
	www.loc.gov /catdir/toc/fy031/2001042958.html (390 words)

	Amazon.com: "Number Cube": Key Phrase page (Site not responding. Last check: 2007-09-08)
		MATH FOR THE TRADES ^ 263 Number Cube Number Cube Number Cube 1 1 36 46,656 71 357,911 2 8 37 50.
		Algebraic Number Theory and Fermat's Last Theorem by Ian Stewart, David Tall
		Magdalene's Lost Legacy: Symbolic Numbers and the Sacred Union in Christianity by Margaret Starbird
	www.amazon.com /phrase/Number-Cube (382 words)

	Publications - Malvina Baica, Ph.D. (Site not responding. Last check: 2007-09-08)
		The Euler System for the Algebraic Number Theory and Mathematical Models In Pollution
		9) Hilbert's demand for the disclosure of units in algebraic number fields
		18) Baica's solution of Fermat's Last Theorem in Euclidean, models and algorithms the transition from abstract to applied mathematics
	math.uww.edu /~baicam/pubs/publications.html (544 words)

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