Lagrange's four-square theorem
Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as a sum of four non-negative integer squares.[1] That is, the squares form an additive basis of order four.
This theorem was proven by
Historical development
From examples given in the
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 k and m. Later, in 1834,
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.[3]
Proofs
The classical proof
Several very similar modern versions[4][5][6] 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.
It is sufficient to prove the theorem for every odd prime number p. This immediately follows from Euler's four-square identity (and from the fact that the theorem is true for the numbers 1 and 2).
The residues of a2 modulo p are distinct for every a between 0 and (p − 1)/2 (inclusive). To see this, take some a and define c as a2 mod p. a is a root of the polynomial x2 − c over the field Z/pZ. So is p − a (which is different from a). In a field K, any polynomial of degree n has at most n distinct roots (Lagrange's theorem (number theory)), so there are no other a with this property, in particular not among 0 to (p − 1)/2.
Similarly, for b taking integral values between 0 and (p − 1)/2 (inclusive), the −b2 − 1 are distinct. By the pigeonhole principle, there are a and b in this range, for which a2 and −b2 − 1 are congruent modulo p, that is for which
Now let m be the smallest positive integer such that mp is the sum of four squares, x12 + x22 + x32 + x42 (we have just shown that there is some m (namely n) with this property, so there is a least one m, and it is smaller than p). We show by contradiction that m equals 1: supposing it is not the case, we prove the existence of a positive integer r less than m, for which rp is also the sum of four squares (this is in the spirit of the infinite descent[7] method of Fermat).
For this purpose, we consider for each xi the yi which is in the same residue class modulo m and between (–m + 1)/2 and m/2 (possibly included). It follows that y12 + y22 + y32 + y42 = mr, for some strictly positive integer r less than m.
Finally, another appeal to Euler's four-square identity shows that mpmr = z12 + z22 + z32 + z42. But the fact that each xi is congruent to its corresponding yi implies that all of the zi are divisible by m. Indeed,
It follows that, for wi = zi/m, w12 + w22 + w32 + w42 = rp, and this is in contradiction with the minimality of m.
In the descent above, we must rule out both the case y1 = y2 = y3 = y4 = m/2 (which would give r = m and no descent), and also the case y1 = y2 = y3 = y4 = 0 (which would give r = 0 rather than strictly positive). For both of those cases, one can check that mp = x12 + x22 + x32 + x42 would be a multiple of m2, contradicting the fact that p is a prime greater than m.
Proof using the Hurwitz integers
Another way to prove the theorem relies on Hurwitz quaternions, which are the analog of integers for quaternions.[8]
The Hurwitz quaternions consist of all quaternions with integer components and all quaternions with half-integer components. These two sets can be combined into a single formula
The (arithmetic, or field) norm of a rational quaternion is the nonnegative rational number
Since quaternion multiplication is associative, and real numbers commute with other quaternions, the norm of a product of quaternions equals the product of the norms:
For any , . It follows easily that is a unit in the ring of Hurwitz quaternions if and only if .
The proof of the main theorem begins by reduction to the case of prime numbers. Euler's four-square identity implies that if Lagrange's four-square theorem holds for two numbers, it holds for the product of the two numbers. Since any natural number can be factored into powers of primes, it suffices to prove the theorem for prime numbers. It is true for . To show this for an odd prime integer p, represent it as a quaternion and assume for now (as we shall show later) that it is not a Hurwitz irreducible; that is, it can be factored into two non-unit Hurwitz quaternions
The norms of are integers such that
If it happens that the chosen has half-integer coefficients, it can be replaced by another Hurwitz quaternion. Choose in such a way that has even integer coefficients. Then
Since has even integer coefficients, will have integer coefficients and can be used instead of the original to give a representation of p as the sum of four squares.
As for showing that p is not a Hurwitz irreducible,
The number u can be factored in Hurwitz quaternions:
The norm on Hurwitz quaternions satisfies a form of the Euclidean property: for any quaternion with rational coefficients we can choose a Hurwitz quaternion so that by first choosing so that and then so that for . Then we obtain
It follows that for any Hurwitz quaternions with , there exists a Hurwitz quaternion such that
The ring H of Hurwitz quaternions is not commutative, hence it is not an actual Euclidean domain, and it does not have unique factorization in the usual sense. Nevertheless, the property above implies that every right ideal is principal. Thus, there is a Hurwitz quaternion such that
In particular, for some Hurwitz quaternion . If were a unit, would be a multiple of p, however this is impossible as is not a Hurwitz quaternion for . Similarly, if were a unit, we would have
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 numbers , can we solve
for all positive integers n in integers ? The case is answered in the positive by Lagrange's four-square theorem. The general solution was given by
Algorithms
In 1986,
Number of representations
The number of representations of a natural number n as the sum of four squares of integers is denoted by r4(n). Jacobi's four-square theorem states that this 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 (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
Some values of r4(n) occur infinitely often as r4(n) = r4(2mn) whenever n is even. The values of r4(n)/n can be arbitrarily large: indeed, r4(n)/n is infinitely often larger than 8√log n.[13]
Uniqueness
The sequence of positive integers which have only one representation as a sum of four squares of non-negative integers (up to order) is:
- 1, 2, 3, 5, 6, 7, 8, 11, 14, 15, 23, 24, 32, 56, 96, 128, 224, 384, 512, 896 ... (sequence A006431 in the OEIS).
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 ... (sequence A000534 in the OEIS).
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,
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 S with such that every positive integer smaller than or equal n can be written as a sum of at most 4 elements of S.[15]
See also
- Fermat's theorem on sums of two squares
- Fermat's polygonal number theorem
- Waring's problem
- Legendre's three-square theorem
- Sum of two squares theorem
- Sum of squares function
- 15 and 290 theorems
Notes
- ISBN 0-486-68252-8
- ^ Ireland & Rosen 1990.
- ^ Sarnak 2013.
- ^ Landau 1958, Theorems 166 to 169.
- ^ Hardy & Wright 2008, Theorem 369.
- ^ Niven & Zuckerman 1960, paragraph 5.7.
- ^ Here the argument is a direct proof by contradiction. With the initial assumption that m > 2, m < p, is some integer such that mp is the sum of four squares (not necessarily the smallest), the argument could be modified to become an infinite descent argument in the spirit of Fermat.
- ^ a b Stillwell 2003, pp. 138–157.
- ^ Ramanujan 1917.
- ^ Oh 2000.
- ^ Rabin & Shallit 1986.
- ^ Pollack & Treviño 2018.
- ^ a b Williams 2011, p. 119.
- ^ Sun 2017.
- ^ Spencer 1996.
References
- ISBN 978-0-19-921985-8.
- Ireland, Kenneth; Rosen, Michael (1990). A Classical Introduction to Modern Number Theory (2nd ed.). Springer. ISBN 978-1-4419-3094-1.
- Landau, Edmund (1958) [1927]. Elementary Number Theory. Vol. 125. Translated by Goodman, Jacob E. (2nd ed.). AMS Chelsea Publishing.
- Niven, Ivan; Zuckerman, Herbert S. (1960). An introduction to the theory of numbers. Wiley.
- Oh, Byeong-Kweon (2000). "Representations of Binary Forms by Quinary Quadratic Forms" (PDF). Trends in Mathematics. 3 (1): 102–107.
- .
- Ramanujan, S. (1917). "On the expression of a number in the form ax2 + by2 + cz2 + dw2". Proc. Camb. Phil. Soc. 19: 11–21.
- Sarnak, Peter (2013). "The Ramanujan Conjecture and some Diophantine Equations". YouTube (Lecture at Tata Institute of Fundamental Research). ICTS Lecture Series. Bangalore, India.
- Zbl 1112.11002.
- S2CID 119597024.
- Williams, Kenneth S. (2011). Number theory in the spirit of Liouville. London Mathematical Society Student Texts. Vol. 76. Zbl 1227.11002.
- Spencer, Joel (1996). "Four Squares with Few Squares". Number Theory: New York Seminar 1991–1995. Springer US. pp. 295–297. ISBN 9780387948263.
- Pollack, P.; Treviño, E. (2018). "Finding the four squares in Lagrange's theorem" (PDF). Integers. 18A: A15.