Exploring the Galois Universe


How to Explore

Let's use gcc and gmp!

1 Download
real quadratic(64-bit)bcn64.c img quadratic (64-bit)bcm64.c
real quadratic (32-bit)bcn32.c img quadratic (32-bit)bcm32.c

2 Edit
Change pn1, pn2 and fn as you like. p should be less than 160 million (64-bit cpu).
Prime-(Mem+Swap): 10M-1.5G, 20M-3G, 40M-6G, 80M-12G, 160M-15G.
f0 and 2*deg*gap should be less than 2^31 (32-bit cpu).

3 Compile and Execute
# gcc -O2 -m64 -o bcn64 bcn64.c -L /usr/local/lib/ -lgmp
# limit stacksize unlimited (or ***M)
# nohup ./bcn64 &

# gcc -O2 -o bcn32 bcn32.c -L /usr/local/lib/ -lgmp
# limit stacksize unlimited (or ***M)
# nohup ./bcn32 &


|f|<200, p<1,000,000 (double-checked)
real realx
img imgx
bcn1.txt: exceptional pairs
[prime p, index k, discriminant d, type]
bcn2.txt: p divides discriminant
bcn3.txt: irregular pairs
[prime p, index k, discriminant d, (root/prime) mod p, type]
f=1,5,8, p<20,000,000 (I am exploring up to 30,000,000.)
real1 real5 real8
[prime, index, discriminant, type]
f=-3, -4, -7, -8, p<20,000,000 (I am exploring up to 30,000,000.)
img3 img4 img7 img8
Exceptional (They are dedicated to Euler.)
34301 and 157229
real1e real5e real8e (34301, 157229)
img3e img4e (379) img7e img8e
Excel file

Check by Gauss sums

[prime p,discriminant d,index k
auxiliary prime l for cyclotomic unit
auxiliary prime l* for gauss sum
ggsum=(gauss sum for l) mod l*
gsum=ggsum^{(l*-1)/p} mod l*]

Euler is the first explorer.
[Imaginary quadratic Gal -200<f<0]
[Real quadratic Gal 0<f<200]

Partially supported by Grant-in-Aid for Young Scientists, (No. 16740019),
the Ministry of Education, Culture, Sports, Science and Technology of Japan.
Partially supported by JSPS KAKENHI Grant Number 21540018, 25400013 and 17K05176.

Reg: 2005, June
Title: Exploring the Galois Universe
EP: 379