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$$\frac {{C+Di}}{{A + Bi}} \equiv \frac {{c + di}}{{a + bi}} \pmod m$$ $$\frac {{B+Di}}{{A + Ci}} \equiv \frac {{\beta + \delta i}}{{\alpha + \gamma i}} \pmod m$$

• Since the correctness of Gauss's first congruence was confirmed (for prime $m$), the remaining question is - what was his motivation?
• How to prove Gauss's second congruence? in this case the trick of writing (for example) $q_3=(A+Bi)+(C+Di)j$ is not adequate to prove it, since the two pairs $A+Ci$ and $B+Di$ are not in their original order in $q_3$.

$$\frac {{C+Di}}{{A + Bi}} \equiv \frac {{c + di}}{{a + bi}} \pmod m$$ $$\frac {{B+Ci}}{{A + Di}} \equiv \frac {{b + ci}}{{a + di}} \pmod m$$

• Since the correctness of Gauss's first congruence was confirmed (for prime $m$), the remaining question is - what was his motivation?
• How to prove Gauss's second congruence? in this case the trick of writing (for example) $q_3=(A+Bi)+(C+Di)j$ is not adequate to prove it, since the two pairs $A+Di$ and $B+Ci$ are not in their original order in $q_3$.

Modern claims that Gauss anticipated the quaternions algebra are based primarily on an unpublished fragment of Gauss dated to 1819 and entitled "rotations of space". In this fragment Gauss describes the quaternion rule for multiplication of two quadrupoles of numbers (which he calls "scales"), remarks it is non-commutative (he does not introduce special notation like $i,j,k$), and gives many formulas that relate 3D spatial rotations to unit quaternions, including one complicated $3\times 3$ orthogonal matrix that acts on a cartesian system $XYZ$ as a rotation. This is mentioned just in order to give background on this fragment of Gauss.

Gauss's congruence

$$\frac {{C+Di}}{{A + Bi}} \equiv \frac {{c + di}}{{a + bi}} \pmod m$$

(actually he writes down six such congruences, but they are all similar in structure so there is no need to write down all of them). For prime $m$, Gauss's congruence is in fact correct, as I will show here.

Proof of Gauss's congruence

Gauss does not explain anything about those six congruences; he just lists them down. Yet it is still surprising that the congruences are in fact correct, even though only for prime $m$. I cannot figure out what was his aim in "mixing" congruences with quaternions, but even if he had an idea here, I think this formula is a very cumbersome way of presenting it. Since the correctness of Gauss's congruence was confirmed (for prime $m$), the remaining question is - what was his motivation?

Example: My unsuccesful interpretation

Modern claims that Gauss anticipated the quaternions algebra are based primarily on an unpublished fragment of Gauss dated to 1819 and entitled "rotations of space". In this fragment Gauss describes the quaternion rule for multiplication of two quadrupoles of numbers (which he calls "scales"), remarks it is non-commutative (he does not introduce special notation like $i,j,k$), and gives many formulas that relate 3D spatial rotations to unit quaternions, including one complicated $3\times 3$ orthogonal matrix that acts on a cartesian system $XYZ$ as a rotation. This is mentioned just in order to give background on this fragment of Gauss.

Gauss's congruences

$$\frac {{C+Di}}{{A + Bi}} \equiv \frac {{c + di}}{{a + bi}} \pmod m$$ $$\frac {{B+Di}}{{A + Ci}} \equiv \frac {{\beta + \delta i}}{{\alpha + \gamma i}} \pmod m$$

(actually he writes down six such congruences, but they are all similar in structure so there is no need to write down all of them). For prime $m$, Gauss's first congruence is in fact correct, as I will show here.

Proof of Gauss's first congruence

Gauss does not explain anything about those six congruences; he just lists them down. Yet it is still surprising that the congruences are in fact correct, even though only for prime $m$. I cannot figure out what was his aim in "mixing" congruences with quaternions, but even if he had an idea here, I think this formula is a very cumbersome way of presenting it.

• Since the correctness of Gauss's first congruence was confirmed (for prime $m$), the remaining question is - what was his motivation?
• How to prove Gauss's second congruence? in this case the trick of writing (for example) $q_3=(A+Bi)+(C+Di)j$ is not adequate to prove it, since the two pairs $A+Ci$ and $B+Di$ are not in their original order in $q_3$.

My unsuccesful interpretation

As far as I understand mathematical intuition, writing down such a formula without "walking" through the usual technical road to it must be the result of some idea, especially in mathematical areas that are considered frontiers (and quaternions were indeed a a frontier of mathematics at the times of Gauss).

As far as I understand mathematical intuition, writing down such a formula without "walking" through the usual technical road to it must be the result of some idea, especially in mathematical areas that are considered frontiers (and quaternions were indeed a frontier of mathematics at the times of Gauss).

Example: My unsuccesful interpretation

I mention this unsuccesful attempt because maybe someone will be able to continue this line of thought and rewrite it in the language of modern abtract algebra.

The proof above views quaternions as ordered pairs of complex numbers (for example: $q_1 = x+jy$ where $x,y$ are complex numbers); Gauss has already encountered the phenomenon of quaternionic behaviour in ordered pairs of complex numbers in another place (Gauss's werke, vol 3, p.384). Therefore I have an intuitive feeling he casted his statements in this particular form because he viewed the quaternions as an hypercomplex number system over the complex numbers (as an extension of $\mathbb{C}$), in the same way complex numbers can be viewed as an hyperreal number system over the reals.

Therefore I thought it is logical to try and search for a similar pattern in the complex numbers. Let $$c_1=a+bi,c_2=\alpha+\beta i$$ $$c_3 = c_1\cdot c_2 =A+Bi = (a\alpha - b\beta)+(b\alpha+a\beta)i$$

and then one gets that $$\frac{B}{A}-\frac{b}{a} = \frac{(b\alpha+a\beta)a-(a\alpha - b\beta)b}{Aa} = \frac{(a^2+b^2)\beta}{Aa} \equiv 0 \pmod {a^2+b^2}$$

where the last relation holds for prime values of $a^2+b^2$. The fact that this congruence is also correct for complex numbers is obvious from the proof for general quaternions, since the complex integers are a subset of quaternion integers. But maybe by looking at Gauss's congruence in the simpler case of $\mathbb{C}$ one can get idea about the kind of notions Gauss attempted to generalize from $\mathbb{C}$ to the quaternions.