  Context. Slowly pulsating B (SPB) stars are main-sequence multi-periodic
oscillators that display non-radial gravity modes. For a fraction of these
pulsators, 4-year photometric light curves obtained with the Kepler space
telescope reveal period spacing patterns from which their internal rotation and
mixing can be inferred. In this inference, any direct resonant mode coupling
has usually been ignored so far.
  Aims. We re-analysed the light curves of a sample of 38 known Kepler SPB
stars. For 26 of those, the internal structure, including rotation and mixing,
was recently inferred from their dipole prograde oscillation modes. Our aim is
to detect direct nonlinear resonant mode coupling among the largest-amplitude
gravity modes.
  Methods. We extract up to 200 periodic signals per star with five different
iterative prewhitening strategies based on linear and nonlinear regression
applied to the light curves. We then identify candidate coupled gravity modes
by verifying whether they fulfil resonant phase relations.
  Results. For 32 of 38 SPB stars we find at least 1 candidate resonance that
is detected in both the linear and the best nonlinear regression model fit to
the light curve and involves at least one of the two largest-amplitude modes.
  Conclusions. The majority of the Kepler SPB stars reveal direct nonlinear
resonances based on the largest-amplitude modes. These stars are thus prime
targets for nonlinear asteroseismic modelling of intermediate-mass dwarfs to
assess the importance of mode couplings in probing their internal physics.
