Gauss’s Last Entry

It is well known that Gauss recorded many of his discoveries in a diary; it ends with the ‘Last Entry’ from July 9, 1814, which reads as follows

1

(see [Kle]):

Observatio per inductionem facta gravissima theoriam residuorum biquadraticorum cum functionibus lemniscaticis elegantissime nectens. Puta si

a

+

bi

est numerus primus,

a

− 1 +

bi

per 2 + 2

i

divisibilis, multitudo omnium solutionum congruentiae

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$$1 \equiv xx + yy + xxyy(\bmod {\text{ }}a + bi) $$

inclusis

x

= ∞,

y

= ±

i;

x

= ±

i

,

y

= ∞ fit = (

a

− 1)

2

+

b

2

.