Fermat polygonal number theorem
Encyclopedia
In additive number theory
, the Fermat polygonal number theorem states that every positive integer is a sum of at most -gonal numbers. That is, every positive number can be written as the sum of three or fewer triangular number
s, and as the sum of four or fewer square number
s, and as the sum of five or fewer pentagonal number
s, and so on. Three such representations of the number 17, for example, are shown below:
The theorem is named after Pierre de Fermat
, who stated it 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' result is sometimes known as the Eureka theorem. The full polygonal number theorem was not resolved until it was finally proven by Cauchy in 1813. The proof of is based on the following lemma due to Cauchy:
For odd positive integers and such that and we can find nonnegative integers , , , and such that
and .
Additive number theory
In number theory, the specialty additive number theory studies subsets of integers and their behavior under addition. More abstractly, the field of "additive number theory" includes the study of Abelian groups and commutative semigroups with an operation of addition. Additive number theory has...
, the Fermat polygonal number theorem states that every positive integer is a sum of at most -gonal numbers. That is, every positive number can be written as the sum of three or fewer triangular number
Triangular number
A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...
s, and as the sum of four or fewer square number
Square number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
s, and as the sum of five or fewer pentagonal number
Pentagonal number
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical...
s, and so on. 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).
The theorem is named after Pierre de Fermat
Pierre de Fermat
Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...
, who stated it without proof, promising to write it in a separate work that never appeared.
Joseph Louis Lagrange
Joseph Louis Lagrange
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...
proved the square case
Lagrange's four-square theorem
Lagrange's four-square theorem, also known as Bachet's conjecture, states that any natural number can be represented as the sum of four integer squaresp = a_0^2 + a_1^2 + a_2^2 + a_3^2\ where the four numbers are integers...
in 1770, which states that every positive number can be represented as a sum of four squares, for example, .. Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
proved the triangular case in 1796, commemorating the occasion by writing in his diary
Gauss's diary
Gauss's diary was a record of the mathematical discoveries of C. F. Gauss from 1796 to 1814. It was rediscovered in 1897 and published by , and reprinted in volume X1 of his collected works.-Sample entries:...
the line "ΕΥΡΗΚΑ!
Eureka (word)
"Eureka" is an interjection used to celebrate a discovery, a transliteration of a word attributed to Archimedes.-Etymology:The word comes from ancient Greek εὕρηκα heúrēka "I have found ", which is the 1st person singular perfect indicative active of the verb heuriskō "I find"...
", and published a proof in his book Disquisitiones Arithmeticae
Disquisitiones Arithmeticae
The Disquisitiones Arithmeticae is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24...
. For this reason, Gauss' result is sometimes known as the Eureka theorem. The full polygonal number theorem was not resolved until it was finally proven by Cauchy in 1813. The proof of is based on the following lemma due to Cauchy:
For odd positive integers and such that and we can find nonnegative integers , , , and such that
and .