Lemniscata geometrice in quinque partes dividitur.
The young Norwegian mathematician Abel published in 1827 and 1828
proofs of the assertions implicit in Gauss's remark in the Disquisitiones. For
m = 3, 5 the necessary constructions had been found much earlier, by 1750,
by the Italian geometer Count Fagnano
The connections of the lemniscate with number theory are too rami ed
for us to discuss them in any more detail. The division points were important
then and are important now, but so are what are called congruences modulo a
prime. The two topics are very closely related. I end with the entry to which
Weil alluded, the very last in the diary, dated July 9, 1814.
Observatio per inducti onem facta gravissima theoriam
residuorum
bi quadraticorum cum functionibus lemniscatis elegantissime nectens.
Puta si a+bi est numerus primus, a-1+bi per 2+2i divisibilis,
multitudo omnium solutionum congruentiae
inclusis
fit
We have progressed a little beyond this last entry in the
intervening 185
years, but not so much as one might think! Without going into further detail,
which would be too much of a digression, I observe that if
and
then
