Exploring the Galois Universe
A new exceptional pair (28679999,2284521) for d=-8 with lmd=2
(2024, January 6)
Parings of p-units in the 4p-cyclotomic field
Programs
cup14-64.c + bcn3.txt -> (4.1) relation3x.txt, (4.2) relation3w.txt
cup14x-64.c + bcn4.txt -> (4.3) relation4.txt
analnew1.c + relation**all.txt -> zero**.txt + irreg**.txt + index**.txt
Data
Relations : (4.1) relation3xall.txt, (4.2) arelation3wall.txt, (4.3) relation4all.txt (each 280MB)
Distribution of the number of zeros: (4.1) zero3x.txt, (4.2) zero3w.txt, (4.3) zero4.txt
Indices of zeros: (4.1) index3x.txt, (4.2) index3w.txt, (4.3) index4.txt
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 &
DATA
|d|<200, p<1,000,000 (double-checked)
real100
real100x
img100
img100x
bcn*.txt
bcn1.txt: exceptional pairs
[prime p, index k, conductor f, type]
bcn2.txt: p divides conductor
bcn3.txt: irregular pairs
[prime p, index k, conductor f, (root/prime) mod p, type]
d=1,5,8, p<30,000,000 (I am exploring up to 40,000,000.)
real1
real5
real8
d=-3, -4, -7, -8, p<30,000,000 (I am exploring up to 40,000,000.)
img3
img4
img7
img8
Exceptional (They are dedicated to Euler.)
379
34301 and 157229
real1e
real5e
real8e (34301, 157229)
img3e
img4e (379)
img7e
img8e New!
Excel file 1(|d|<200, p<1,000,000)
Excel file 2(|d|<10, p<20,000,000)
Check of exceptional pairs on v_p(a_0), v_p(b_0) and lambda^- by UBASIC
programs and results
programs: IA7SS.UB (real), IA7SSIM.UB (img)
list of exceptional pairs: ia7ss.txt (real), ia7ssim.txt (img)
[prime p, index k, conductor f]
results: IA7SS.UBD (real), IA7SSIM.UBD (img)
[conductor f, prime p, index k
lambda: Coeff(0), Coeff(1), Coeff(2) mod p
v_p(b_0)=e*: Coeff(0) mod p^2, Coeff(0) mod p^3
v_p(a_0)=e: Coeff'(0) mod p^2, Coeff'(0) mod p^3 ]
Check of exceptional pairs on nu^+ by Gauss sums
programs and results
programs: gauss64.c (small), gauss64d.c (large)
results: g-sum1.txt, g-sum2.txt (small), g-sum3.txt, g-sum4.txt (large)
[prime p, conductor f, 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.
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, 17K05176 and JP20H00115.
Reg: 2005, June
Title: Exploring the Galois Universe
Method:webpage
Content:
https://math0.pm.tokushima-u.ac.jp/~hiroki/major/galois1-e.html
EP: 379
Index(English)
Index(Japanese)