2000 | OriginalPaper | Chapter
Gauss’s Last Entry
Author : Franz Lemmermeyer
Published in: Reciprocity Laws
Publisher: Springer Berlin Heidelberg
Included in: Professional Book Archive
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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
% MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaaIXaGaeyyyIORaamiEaiaadIhacqGHRaWkcaWG5bGaamyEaiab % gUcaRiaadIhacaWG4bGaamyEaiaadMhacaGGOaWdaiaaykY7peGaci % yBaiaac+gacaGGKbWdaiaaykY7caqGGaWdbiaadggacqGHRaWkcaWG % IbGaamyAaiaacMcaaaa!4E3C!
$$1 \equiv xx + yy + xxyy(\bmod {\text{ }}a + bi) $$
inclusis
x
= ∞,
y
= ±
i;
x
= ±
i
,
y
= ∞ fit = (
a
− 1)
2
+
b
2
.