We apply noise thermometry to characterize charge and thermoelectric transport in single InAs nanowires (NWs) at a bath temperature of 4.2 K. Shot noise measurements identify elastic diffusive transport in our NWs with negligible electron-phonon interaction. This enables us to set up a measurement of the diffusion thermopower. Unlike in previous approaches, we make use of a primary electronic noise thermometry to calibrate a thermal bias across the NW. In particular, this enables us to apply a contact heating scheme, which is much more efficient in creating the thermal bias as compared to conventional substrate heating. The measured thermoelectric Seebeck coefficient exhibits strong mesoscopic fluctuations in dependence on the back-gate voltage that is used to tune the NW carrier density. We analyze the transport and thermoelectric data in terms of approximate Mott's thermopower relation and to evaluate a gate-voltage to Fermi energy conversion factor.Efficient thermoelectric (TE) conversion in solid state devices has been an elusive target for many decades. An ideal TE material should display a large electrical conductivity σ and Seebeck coefficient S, and a small heat conductivity κ 1 . On the other hand, bulk materials are typically characterized by a strong interdependence between these parameters, which poses limits to the maximum achievable conversion efficiency 1 . Nanostructured semiconductors today offer a host of novel ways to elude part of these constraints and are leading to a promising new direction in TE research 2-4 . For instance, present evidences show that phonon conductivity can be significantly suppressed in nanostructures 5-9 and promising results have also been obtained on the tuning of the TE response through an engineering of electron quantum states 10-12 . The investigation of TE effects in nanoscale conductors, though, brings with it a set of technical challenges linked to reproducibility and accuracy in the estimate of the TE properties of single nanostructures. In particular, electrical and heat contact resistances 8,9 are often difficult to predict and measure, as well as the relative impact of the different transport mechanisms in the emergence of the nanomaterials TE properties. In addition, the role of phonons and of their interaction with the electron system is often hard to access in a real nanostructure 13 . This calls for novel measurement methods to correlate various aspects of the TE response of a nanomaterial and sort out the fundamental physics ruling their TE behavior 14,15 .In our work, we investigate the TE response of individual InAs NWs at a temperature of few Kelvins and demonstrate a primary thermometry method based on current noise measurements. Our approach allows first of all a direct measurement of the thermal bias across the device, and covers a fairly large operation temperature range going well beyond the one typically available with superconductive tunnel junctions 16 . In addition, weshow that the investigation of shot noise as a function of the bias offers...