Circularity in finite fields and solutions of the equations x ^m  +  y ^m  −  z ^m  = 1

In this article, we investigate 2-$(v,k,\lambda )$ designs with $gcd(r,\lambda )=1$ admitting flag-transitive automorphism groups G. We prove that if G is an almost simple group, then such a design belongs to one of the eight infinite families of 2-designs or it is ...$\mathcal{}$$$$\mathrm{}$$\mathrm{}\mathrm{}{\mathrm{}}_{\mathrm{}}$$\mathcal{}$

Let @a,@b@?F"q"^"t^* and let N"t(@a,@b) denote the number of solutions (x,y)@?F"q"^"t^*xF"q"^"t^* of the equation x^q^-^1+@ay^q^-^1=@b. Recently, Moisio determined N"2(@a,@b) and evaluated N"3(@a,@b) in terms of the number of rational points on a ...

This paper is a contribution to the investigation of flag-transitive non-symmetric 2-designs. We prove that if $${\mathcal {D}}$$D is a non-trivial non-symmetric 2-$$(v,k,\lambda )$$(v,k,ź) design with $$(r,\lambda )=1$$(r,ź)=1 and $$G\le Aut({\mathcal {...