Solutions of the equations y² = xⁿ+k (n = 3,4) in a finite field are given almost explicitly in terms of k.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba52-3-1, author = {A. Schinzel and M. Ska\l ba}, title = {On Equations y2 = xn+k in a Finite Field}, journal = {Bulletin of the Polish Academy of Sciences. Mathematics}, volume = {52}, year = {2004}, pages = {223-226}, zbl = {1105.11015}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba52-3-1} }
A. Schinzel; M. Skałba. On Equations y² = xⁿ+k in a Finite Field. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 52 (2004) pp. 223-226. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba52-3-1/