Lagrange's four-square theorem, also known as Bachet's conjecture, states that every
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
can be represented as the sum of four integer
squares. That is, the squares 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 o ...
of order four.
where the four numbers
are integers. For illustration, 3, 31, and 310 in several ways, can be represented as the sum of four squares as follows:
This theorem was proven by
Joseph Louis Lagrange in 1770. It is a special case of the
Fermat polygonal number theorem.
Historical development
From examples given in the ''
Arithmetica
''Arithmetica'' ( grc-gre, Ἀριθμητικά) is an Ancient Greek text on mathematics written by the mathematician Diophantus () in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate ...
,'' it is clear that
Diophantus
Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...
was aware of the theorem. This book was translated in 1621 into Latin by
Bachet (Claude Gaspard Bachet de Méziriac), who stated the theorem in the notes of his translation. But the theorem was not proved until 1770 by Lagrange.
Adrien-Marie Legendre extended the theorem in 1797–8 with his
three-square theorem, by proving that a positive integer can be expressed as the sum of three squares if and only if it is not of the form
for integers and . Later, in 1834,
Carl Gustav Jakob Jacobi discovered a simple formula for the number of representations of an integer as the sum of four squares with his own
four-square theorem.
The formula is also linked to
Descartes' theorem of four "kissing circles", which involves the sum of the squares of the curvatures of four circles. This is also linked to
Apollonian gaskets, which were more recently related to the
Ramanujan–Petersson conjecture In mathematics, the Ramanujan conjecture, due to , states that Ramanujan's tau function given by the Fourier coefficients of the cusp form of weight
:\Delta(z)= \sum_\tau(n)q^n=q\prod_\left (1-q^n \right)^ = q-24q^2+252q^3- 1472q^4 + 4830q^5-\c ...
.
Proofs
The classical proof
Several very similar modern versions of Lagrange's proof exist. The proof below is a slightly simplified version, in which the cases for which ''m'' is even or odd do not require separate arguments.
Proof using the Hurwitz integers
Another way to prove the theorem relies on
Hurwitz quaternions, which are the analog of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s for
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s.
[.]
Generalizations
Lagrange's four-square theorem is a special case of the
Fermat polygonal number theorem and
Waring's problem. Another possible generalization is the following problem: Given
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s
, can we solve
for all positive integers in integers
? The case
is answered in the positive by Lagrange's four-square theorem. The general solution was given by
Ramanujan. He proved that if we assume, without loss of generality, that
then there are exactly 54 possible choices for
such that the problem is solvable in integers
for all . (Ramanujan listed a 55th possibility
, but in this case the problem is not solvable if
.)
Algorithms
In 1986,
Michael O. Rabin and
Jeffrey Shallit proposed
randomized
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual ran ...
polynomial-time algorithms for computing a single representation
for a given integer , in expected running time
. It was further improved to
by Paul Pollack and Enrique Treviño in 2018.
Number of representations
The number of representations of a natural number ''n'' as the sum of four squares is denoted by ''r''
4(''n'').
Jacobi's four-square theorem states that this is eight times the sum of the
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s of ''n'' if ''n'' is odd and 24 times the sum of the odd divisors of ''n'' if ''n'' is even (see
divisor function), i.e.
Equivalently, it is eight times the sum of all its divisors which are not divisible by 4, i.e.
We may also write this as
where the second term is to be taken as zero if ''n'' is not divisible by 4. In particular, for a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
''p'' we have the explicit formula .
[.]
Some values of ''r''
4(''n'') occur infinitely often as whenever ''n'' is even. The values of ''r''
4(''n'')/''n'' can be arbitrarily large: indeed, ''r''
4(''n'')/''n'' is infinitely often larger than 8.
Uniqueness
The sequence of positive integers which have only one representation as a sum of four squares (up to order) is:
:1, 2, 3, 5, 6, 7, 8, 11, 14, 15, 23, 24, 32, 56, 96, 128, 224, 384, 512, 896 ... .
These integers consist of the seven odd numbers 1, 3, 5, 7, 11, 15, 23 and all numbers of the form
or
.
The sequence of positive integers which cannot be represented as a sum of four ''non-zero'' squares is:
:1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 24, 29, 32, 41, 56, 96, 128, 224, 384, 512, 896 ... .
These integers consist of the eight odd numbers 1, 3, 5, 9, 11, 17, 29, 41 and all numbers of the form
or
.
Further refinements
Lagrange's four-square theorem can be refined in various ways. For example,
Zhi-Wei Sun
Sun Zhiwei (, born October 16, 1965) is a Chinese mathematician, working primarily in number theory, combinatorics, and group theory. He is a professor at Nanjing University.
Biography
Sun Zhiwei was born in Huai'an, Jiangsu. Sun and his ...
proved that each natural number can be written as a sum of four squares with some requirements on the choice of these four numbers.
One may also wonder whether it is necessary to use the entire set of square integers to write each natural as the sum of four squares.
Eduard Wirsing proved that there exists a set of squares with
such that every positive integer smaller than or equal can be written as a sum of at most 4 elements of .
[Spencer 1996.]
See also
*
Fermat's theorem on sums of two squares
*
Fermat's polygonal number theorem
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 integer can be written as the sum of three or fewer triangular numbers, and as the sum ...
*
Waring's problem
*
Legendre's three-square theorem
*
Sum of two squares theorem
In number theory, the sum of two squares theorem relates the prime decomposition of any integer to whether it can be written as a sum of two squares, such that for some integers , .
:''An integer greater than one can be written as a sum of t ...
*
Sum of squares function
*
15 and 290 theorems
Notes
References
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External links
Proof at PlanetMath.orgAnother proofan applet decomposing numbers as sums of four squaresOEIS index to sequences related to sums of squares and sums of cubes
{{DEFAULTSORT:Lagrange's Four-Square Theorem
Additive number theory
Articles containing proofs
Squares in number theory
Theorems in number theory