additive number theory
Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigr ...
, the Fermat polygonal number theorem states that every positive integer is a sum of at most
-gonal number In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers.
Definition and examples
...
s. That is, every positive integer can be written as the sum of three or fewer
triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
s, and as the sum of four or fewer
square number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
s, and as the sum of five or fewer
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. The ...
s, and so on. That is, the -gonal numbers form an
additive basis In additive number theory, an additive basis is a set S of natural numbers with the property that, for some finite number k, every natural number can be expressed as a sum of k or fewer elements of S. That is, the sumset of k copies of S consists of ...
of order .
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
The theorem is named after
Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
, who stated it, in 1638, without proof, promising to write it in a separate work that never appeared..
Joseph Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiasquare case in 1770, which states that every positive number can be represented as a sum of four squares, for example, .
Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
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
The (Latin for "Arithmetical Investigations") 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. It is notable for having had a revolutionary impact on th ...
. For this reason, Gauss's result is sometimes known as the Eureka theorem.. The full polygonal number theorem was not resolved until it was finally proven by
Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
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 .
See also
*
Pollock's conjectures Pollock's conjectures are two closely related unproven conjectures in additive number theory. They were first stated in 1850 by Sir Frederick Pollock, better known as a lawyer and politician, but also a contributor of papers on mathematics to the R ...
*
Waring's problem
In number theory, Waring's problem asks whether each natural number ''k'' has an associated positive integer ''s'' such that every natural number is the sum of at most ''s'' natural numbers raised to the power ''k''. For example, every natural num ...