We consider Thue equations of the form $a{x}^k+b{y}^k=1$, and assuming the truth of the abc-conjecture, we show that almost all locally soluble Thue equations of degree at least three violate the Hasse principle. A similar conclusion holds true for Fermat equations $a{x}^k+b{y}^k+c{z}^k=0$ of degree at least six.

For any positive power n of a prime p we find a complete set of generating relations between the elements [r] = rⁿ - r and p·1 of a unitary commutative ring.