The calculation of $\pi_* TP(\mathbb{F}_p) = \pi_* THH(\mathbb{F}_p)^{tS^1} = \pi_* \widehat{\mathbb{H}}(S^1, THH(\mathbb{F}_p))$ (the notation has changed over the years) was first published by Hesselholt-Madsen (Topology, 1997). The calculation of $\pi_* TP(\mathbb{Z})/p$ for odd primes $p$ was published by B"okstedt-Madsen (conference proceedings, 1994 and 1995) and, independently, by Tsalidis (Amer J. Math, 1997). They also calculated $\pi_* TC(\mathbb{Z})/p$. Comparison with known spectra related to topological $K$-theory allowed a determination of $\pi_* TC(\mathbb{Z})_p$, see Rognes (Math. Proc. Camb. Philos Soc., 1993, Corollary 3). I also made the corresponding calculations of $\pi_* TP(\mathbb{Z})/2$, $\pi_* TC(\mathbb{Z})/2$ and $\pi_* TC(\mathbb{Z})_2$ (Journal of Pure and Applied Algebra, 1999). There is some discussion of the additive extensions in the Tate spectral sequence for $\pi_* TP(\mathbb{Z})_2$ in Theorem 1.9 on page 231 of one of those JPAA papers, but the general picture appears to be complicated. My former PhD student Knut Berg determined (ca. 2013) the continuous mod $2$ homology of $TP(\mathbb{Z})$ as an $A_*$-comodule algebra, where $A_*$ is the dual Steenrod algebra, as well as the Adams $E_2$-term and $d_2$-differentials. This gives some other information about $\pi_* TP(\mathbb{Z})_2$ than the Tate spectral sequence, but a complete picture of the later differential pattern is missing.