**EDIT:** I've tried to alter the question so that its basic nature is clearer, as it's been unclear to a number of people now.


At any prime p, there is a graded polynomial ring $V \cong {\mathbb Z}\_{(p)}[v\_1, v\_2, \ldots]$ carrying two formal group laws. These formal group laws are of the form
$$
F(x,y) = \ell^{-1}(\ell(x) + \ell(y))
$$
for a logarithm $\ell(x) = \sum \ell\_n x^{p^{n+1}} \in ({\mathbb Q} \otimes V)[\![x]\!]$ (where $\ell\_0 = 1$ by convention). *Both* of these formal group laws have the property that they are universal among so-called $p$-typical formal group laws, and see heavy computational use in stable homotopy theory.


These two are based on choices of recursive definition for the logarithm coefficients in terms of the generators $v\_i$ of $V$. The first definition (the Araki generators) satisfies:
$$
p \ell\_n = \sum\_{k=0}^n v\_k^{p^{n-k}} \ell\_k = v\_n + \ell\_1 v\_{n-1}^p + \cdots + \ell\_{n-1} v\_1^{p^{n-1}} + \ell\_n p^{p^n}
$$
The Hazewinkel generators are instead defined by:
$$
p \ell'\_n = \sum\_{k=1}^{n} v\_k^{p^{n-k}} \ell'\_k = v\_n + \ell'\_1 v\_{n-1}^p + \cdots + \ell'\_{n-1} v\_1^{p^{n-1}}
$$
This gives the ring $V$ with two logarithms $\ell$ and $\ell'$, and two distinct universal formal group laws.


My question is: Are these two formal group laws isomorphic? Strictly isomorphic?


**ADDED:** Since the ring is torsion free, any isomorphism between them is of the form $f(x) = (\ell')^{-1} (c \ell(x))$ for a unit $c \in \mathbb{Z}\_{(p)}^\times = V^\times$. They are therefore isomorphic if and only if they are strictly isomorphic. Therefore, the question is equivalent to the following:


**Does the power series $(\ell')^{-1} \circ \ell$ have coefficients in $V \subset V \otimes \mathbb{Q}$?**


(The issue was brought up when thinking about truncated Brown-Peterson spectra ${\rm BP}\langle n\rangle$, whose rings of coefficients are $V/(v\_{n+1},v\_{n+2}, \cdots)$. It then becomes a question as to whether these are equivalent as ring spectra depending on the choice of generators. There are certainly different choices of generators for which they are inequivalent.)



Isomorphism between two universal p-typical formal group laws
No, at least when $p=2$ the coefficient of $x^8$ in the relevant power series is not 2-locally integral. I have put a Maple worksheet at <https://strickland1.org/misc/ArHaz.mw> with a PDF version at <https://strickland1.org/misc/ArHaz.pdf>.


