An elementary evaluation of the Jacobsthal sum
The paper contains a very interesting and elementary new proof of the theorem of Jacobsthal that if \(p\equiv 1 (4)\) is prime so that \(p=a^ 2+b^ 2\), \(a\equiv -1 (4)\) (a unique) and if \((\frac{r}{p})=1\), say \(r\equiv u^ 2 (mod p)\), then \[ S_ r=\sum^{p- 1}_{x=1}(\frac{x(x^ 2+r)}{p})=(\frac{u}{p}) 2a. \] The machinery used is the decomposition of a reduced residue system 1,2,...,p-1 into 'cells' and then defining singletons, right and left end points and interior points, their number being denoted by s, e and i, respectively. The author then proves easy results connecting the quantities s,e,i amongst themselves and with \(S_ r\) and the theorem then emerges rather elegantly ! It would be nice to see some more applications of the quantities s,e,i.
scientific article; zbMATH DE number 3922771