Jacobi's four-square theorem

For every positive integer n, the number of ordered integer quadruples satisfying n = a² + b² + c² + d² is eight times the sum of the positive divisors of n not divisible by four.

Sum-of-divisors function: The arithmetic function that sums the positive divisors of an integer. The theorem uses a restricted divisor sum to count representations.
Theta function: A series that encodes arithmetic or geometric data through powers of a variable. A fourth power of the square theta series records four-square representations.
Carl Gustav Jacob Jacobi: A German mathematician whose work shaped elliptic functions, number theory, and mechanics. He established the four-square identity that bears his name.
Sum of four squares: An expression of an integer as the sum of four integer squares. The theorem gives the exact count for each positive integer.
Four-square representation function: The function counting ordered integer solutions to n = a² + b² + c² + d². It is the quantity Jacobi’s formula evaluates exactly.
Jacobi triple product: An identity equating an infinite product with a bilateral series involving powers of a variable. Product expansions provide identities used in Jacobi’s theta-function calculations.
Disquisitiones Arithmeticae: Carl Friedrich Gauss’s 1801 treatise that systematized major parts of number theory. Its arithmetic tradition frames the problem of representing integers by quadratic forms.
Representation number: The number of solutions expressing an integer through a specified form or equation. Jacobi’s formula is an exact representation-number evaluation.
Ordered tuple: A finite sequence whose entries occupy distinct positions, so rearrangements can differ. The theorem counts permutations as distinct representations.
Modular form: A holomorphic function on the upper half-plane with specified transformation and growth properties. The theorem’s generating series belongs to the theory of modular forms.

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Legendre's three-square theoremRelated: It extends the representation-counting perspective from existence to the number of four-square representations.