Proving two unpublished assertions by Gauss on special values of lemniscatic functions.

P.412 of volume 3 of Gauss's collected works contains two unpublished remarks of Gauss that apparently have not been discussed yet. The first one is of number-theoretic significance, while the second relates the value of $Q([a+bi]\varphi)$ at a point $\varphi=\text{arcsinlemn} (x)$ such that $x^4=\pm 1$, to the norm $M(\pi)=a^2+b^2$. Here $\pi=a+bi$ is a Gaussian integer and $\text{sinlemn} (\varphi)=\frac{P(\varphi)}{Q(\varphi)}$.

Here is a translation of the first statement:

"Among the numbers $y = x, 2x+1, 2x+i, 2x−1, 2x−i$, there is always at least one (or three, or all of them) for which the congruence $1-y^4\equiv z^2$ is solvable with respect to any given modulus."

The second statement is not stated in its most general form, but its precise form can be assumed from the several special cases Gauss mentions:

(here $x=\text{sinlemn}(\varphi)$) \begin{array}{c|c|l} a+bi & M & Q(a+bi)\varphi \\ \hline -1+2i & 5 & 1+(-1+2i)x^4 \\ -3 & 9 & 1+6x^4-3x^8 \\ 3+2i & 13 & 1+(-11+10i)x^4+(7-4i)x^8+(3+2i)x^{12} \\ 1+4i & 17 & 1+(12-20i)x^4+(-10+28i)x^8-(20+12i)x^{12}+(1+4i)x^{16} \\ 5 & 25 & 1+50x^4-125x^8+300x^{12}-105x^{16}-62x^{20}+5x^{24} \ \end{array} For $x^4=+1$ and $x^4=-1$, these functions become respectively the squares and the cubes of $1-i,-2,-2-2i,-4i,+8$ for $M=5,9,13,17,25$.

The general pattern in Gauss's second assertion

The numbers mentioned by Gauss (i.e $1-i,-2,-2-2i,-4i,+8$) are precisely associates of $(1-i)^{\frac{M-1}{4}} = (1-i)^{\frac{a^2+b^2-1}{4}}$; here an associate of a Gaussian integer $c+di$ is $c+di$ times a unit $\in \left\{1,+i,-1,-i\right\}$.

So Gauss's second claim is:

For $x^4=1$ the value of $Q([a+bi]\varphi)$ is an associate of $(1-i)^{\frac{a^2+b^2-1}{2}}$. For $x^4 = -1$ the value of $Q([a+bi]\varphi)$ is an associate of $(1-i)^{\frac{3(a^2+b^2-1)}{4}}$.

Questions

Of course that it will be best if someone can prove these assertions; in any other case, it will be good if someone will be able to place these statements in the right mathematical context! The editors of Gauss's works did not comment on these particular results, so it is very hard for to me to understand its meaning.