  In 1981, Foias, Guillop\'e and Temam proved a priori estimates for
arbitrary-order space derivatives of solutions to the Navier-Stokes equation.
Such bounds are instructive in the numerical investigation of intermittency
often observed in simulations, e.g., numerical study of vorticity moments by
Donzis et al. (2013) revealed depletion of nonlinearity that may be responsible
for smoothness of solutions to the Navier-Stokes equation. We employ an
original method to derive analogous estimates for space derivatives of
three-dimensional space-periodic weak solutions to the evolutionary equations
of diffusive magnetohydrodynamics. Construction relies on space analyticity of
the solutions at almost all times. An auxiliary problem is introduced, and a
Sobolev norm of its solutions bounds from below the size in $C^3$ of the region
of space analyticity of the solutions to the original problem. We recover the
exponents obtained earlier for the hydrodynamic problem. The same approach is
also followed here to derive and prove similar a priori bounds for
arbitrary-order space derivatives of the first-order time derivative of the
weak MHD solutions.
  This paper is dedicated to Professor Uriel Frisch on the occasion of his 80th
anniversary as a sign of appreciation of the Scientist and the Teacher.
