Two representations are considered different if their terms are in different order or if the integer being squared (not just the square) is different; to illustrate, these are three of the eight different ways to represent 1:
The number of ways to represent n as the sum of four squares 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
where the second term is to be taken as zero if n is not divisible by 4. In particular, for a prime numberp we have the explicit formula r4(p) = 8(p + 1).[1]
Some values of r4(n) occur infinitely often as r4(n) = r4(2mn) whenever n is even. The values of r4(n) can be arbitrarily large: indeed, r4(n) is infinitely often larger than [1]
^Hirschhorn, Michael D. (2000). "Partial Fractions and Four Classical Theorems of Number Theory". The American Mathematical Monthly. 107 (3): 260–264. CiteSeerX10.1.1.28.1615. doi:10.2307/2589321. JSTOR 2589321.
Referencesedit
Hirschhorn, Michael D.; McGowan, James A. (2001). "Algebraic Consequences of Jacobi's Two— and Four—Square Theorems". In Garvan, F. G.; Ismail, M. E. H. (eds.). Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Developments in Mathematics. Vol. 4. Springer. pp. 107–132. CiteSeerX10.1.1.26.9028. doi:10.1007/978-1-4613-0257-5_7. ISBN978-1-4020-0101-7.
Hirschhorn, Michael D. (1987). "A simple proof of Jacobi's four-square theorem". Proceedings of the American Mathematical Society. 101 (3): 436. doi:10.1090/s0002-9939-1987-0908644-9.
jacobi, four, square, theorem, other, theorems, jacobi, jacobi, theorem, disambiguation, number, theory, gives, formula, number, ways, that, given, positive, integer, represented, four, squares, integers, contents, history, theorem, proof, also, notes, referen. For other theorems of Jacobi see Jacobi s theorem disambiguation In number theory Jacobi s four square theorem gives a formula for the number of ways that a given positive integer n can be represented as the sum of four squares of integers Contents 1 History 2 Theorem 3 Proof 4 See also 5 Notes 6 References 7 External linksHistory editThe theorem was proved in 1834 by Carl Gustav Jakob Jacobi Theorem editTwo representations are considered different if their terms are in different order or if the integer being squared not just the square is different to illustrate these are three of the eight different ways to represent 1 12 02 02 0202 12 02 02 1 2 02 02 02 displaystyle begin aligned 1 2 amp 0 2 0 2 0 2 0 2 amp 1 2 0 2 0 2 1 2 amp 0 2 0 2 0 2 end aligned nbsp The number of ways to represent n as the sum of four squares 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 r4 n 8 m nmif n is odd 24 m nm oddmif n is even displaystyle r 4 n begin cases displaystyle 8 sum m n m amp text if n text is odd 12pt displaystyle 24 sum m n atop m text odd m amp text if n text is even end cases nbsp Equivalently it is eight times the sum of all its divisors which are not divisible by 4 i e r4 n 8 m n 4 mm displaystyle r 4 n 8 sum m mid n atop 4 nmid m m nbsp We may also write this asr4 n 8s n 32s n 4 displaystyle r 4 n 8 sigma n 32 sigma n 4 nbsp where the second term is to be taken as zero if n is not divisible by 4 In particular for a prime number p we have the explicit formula r4 p 8 p 1 1 Some values of r4 n occur infinitely often as r4 n r4 2mn whenever n is even The values of r4 n can be arbitrarily large indeed r4 n is infinitely often larger than 8log n displaystyle 8 sqrt log n nbsp 1 Proof editThe theorem can be proved by elementary means starting with the Jacobi triple product 2 The proof shows that the Theta series for the lattice Z4 is a modular form of a certain level and hence equals a linear combination of Eisenstein series See also editLagrange s four square theorem Lambert series Sum of squares functionNotes edit a b Williams 2011 p 119 Hirschhorn Michael D 2000 Partial Fractions and Four Classical Theorems of Number Theory The American Mathematical Monthly 107 3 260 264 CiteSeerX 10 1 1 28 1615 doi 10 2307 2589321 JSTOR 2589321 References editHirschhorn Michael D McGowan James A 2001 Algebraic Consequences of Jacobi s Two and Four Square Theorems In Garvan F G Ismail M E H eds Symbolic Computation Number Theory Special Functions Physics and Combinatorics Developments in Mathematics Vol 4 Springer pp 107 132 CiteSeerX 10 1 1 26 9028 doi 10 1007 978 1 4613 0257 5 7 ISBN 978 1 4020 0101 7 Hirschhorn Michael D 1987 A simple proof of Jacobi s four square theorem Proceedings of the American Mathematical Society 101 3 436 doi 10 1090 s0002 9939 1987 0908644 9 Williams Kenneth S 2011 Number theory in the spirit of Liouville London Mathematical Society Student Texts Vol 76 Cambridge University Press ISBN 978 0 521 17562 3 Zbl 1227 11002 External links editWeisstein Eric W Sum of Squares Function MathWorld Retrieved from https en wikipedia org w index php title Jacobi 27s four square theorem amp oldid 1178673355, wikipedia, wiki, book, books, library,
