The primitive solutions of seven small generalized Fermat equations

Version 9 of Grechuk's list of open Diophantine equations contains seven equations ax^p+by^q+cz^r=0 with 1/p+1/q+1/r<1 and |a|2^p+|b|2^q+|c|2^r<=60 for which the complete list of primitive solutions was not known. We determine the primitive solutions of all seven: x^5+y^4-2z^2=0, x^5+y^4-3z^2=0, x^5+y^4+3z^2=0, 2x^4-y^4+z^3=0, x^4+3y^3+2z^3=0, x^4+4y^3+z^3=0 and x^4+3y^3+z^3=0. Besides solutions with entries 0 and ±1, the only ones are (11,±29,±538) of x^5+y^4=3z^2 and (±895,-6202,4199) of x^4+3y^3+z^3=0. For the last equation this answers a question of Grechuk; J. Agbanwa has announced a proof in an unpublished manuscript, and ours is by a different method. The proofs combine descent over number fields of small degree with elliptic Chabauty and the Mordell-Weil sieve. For x^4+3y^3+z^3=0 we also use a covering by a curve of genus one over a number field of degree 12, whose Jacobian has unknown rank, and Stoll's form of Chabauty's method at the prime 2, which uses a Selmer group of a 2-isogeny in place of the Mordell-Weil group. No step assumes the generalized Riemann hypothesis. Programs and data: https://doi.org/10.5281/zenodo.23055562.