2.4.2 if Par

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1 Pari-GP 1 Pari-gp Pari-gp gp gp gp qfbclassno() qfbclassno() quadclassunit() quadclassunit() bnfinit() bnfinit() tech bnfcertify() gp

2 2.4.2 if Pari-gp C Pari GP2C GP C GP2C GP2C Pari-gp Pari-gp[4] free GNU Octave 1 Mathematica Maple Pari-gp gp Pari C gp GP H. Cohen K. Belabas Pari-gp GNU GPL free software CVS Unix Linux FreeBSD, Apple Mac OS X Unix Microsoft Windows cygwin 1 2

3 FreeBSD Linux Microsoft Windows Microsoft Windows cygwin Pari-gp Pari-gp gp 4MB 500, C gp 3

4 gp for() C gp2c gp C 3.4 Abel Smith Normal Form, Abel Hermite Normal Form, H. Cohen [2], [3] GRH GRH Heuristic Pari-gp 2.2.3, gp gp t INT 4

5 t INT t INTMOD t FRAC t REAL t COMPLEX p t PADIC t POL t SER Pari-gp t VEC C hash GP t MAT exact exact Pari-gp GMP 2 p gp gp Unix shell gp Pari-gp 3.2 iwao@jeff:~/docmath/hokurikuntws-2004dec {537}$ gp Reading GPRC: /home/iwao/.gprc...done. GP/PARI CALCULATOR Version (development CHANGES ) i386 running freebsd (ix86/gmp kernel) 32-bit version compiled: Apr , gcc [FreeBSD] (readline v4.3 enabled, extended help available) 2 GNU Multiple Precision Arithmetic Library, 5

6 Copyright (C) The PARI Group PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER. Type? for help, \q to quit. Type?12 for how to get moral (and possibly technical) support. parisize = , primelimit = ? readline v4.3 enabled extended help available dvi? gp \q parisize = , primelimit = ? \q Goodbye! ? 2^100 %2 = ? 1/2 + 1/3 %3 = 5/6? %+ 0.0 %4 = % 6

7 [1,2,3] [1,2,3]~ [1,2,3;4,5,6;7,8,9] ( ) ? a = 1;? a = [1, 2, 3];? a[1] 1 x 2 + 2x + 1? x^2+2*x+1 %5 = x^2 + 2*x factor()? factor(%) %6 = [x + 1 2] x 2 + 2x + 1 = (x + 1) 2 gp 7

8 factor() x, flag x? factor(2^15+1,10) %5 = [3 2] [3641 1]? factor(2^15+1) %6 = [3 2] [11 1] [331 1] gp? gp?? Help topics: for a list of relevant subtopics, type?n for n in 0: user-defined identifiers (variable, alias, function) 1: Standard monadic or dyadic OPERATORS 2: CONVERSIONS and similar elementary functions 3: TRANSCENDENTAL functions 4: NUMBER THEORETICAL functions 5: Functions related to ELLIPTIC CURVES 6: Functions related to general NUMBER FIELDS 7: POLYNOMIALS and power series 8: Vectors, matrices, LINEAR ALGEBRA and sets 9: SUMS, products, integrals and similar functions 10: GRAPHIC functions 11: PROGRAMMING under GP 12: The PARI community gp?4 4 NUMBER THEORETICAL functions??4 addprimes bestappr bezout bezoutres bigomega binomial chinese content contfrac contfracpnqn core coredisc dirdiv direuler dirmul 8

9 divisors eulerphi factor factorback factorcantor factorff factorial factorint factormod ffinit fibonacci gcd hilbert isfundamental ispower isprime ispseudoprime issquare issquarefree kronecker???factor factor(x,{lim}): factorization of x. lim is optional and can be set whenever x is of (possibly recursive) rational type. If lim is set return partial factorization, using primes up to lim (up to primelimit if lim=0)?? TEX dvi file Pari-gp extended help avairable gp quad qfb quaddisc(x) Q( x) qfbclassno() qfbclassno(x) Q( x) Shanks exponent 9

10 2.2.2 qfbclassno()? qfbclassno (-3299) %10 = 27 qfbclassno (-3299,1) %11 = 27 Euler quadclassunit() quadclassunit(x) Q( x) Buchmann-McCarthy sub exponential Abel quadclassunit(x,,tech) 3 tech 2 [c,d] d=b GRH d=6 b quadclassunit()? quadclassunit(-10^25-3) time = 10,810 ms. %20 = [ , [ , 2], [Qfb(7, 1, ), Qfb(13, 13, )], 1] [,,, ]. 2.3 nf bnf big number field 10

11 2.3.1 bnfinit() bnfinit() 1 Buchmann sub exponential algorithm[2, Chap. 6.5] bnfinit() 1 x x x = 0 K? T = x^4 + 24*x^ *x ;? bnf = bnfinit(t); bnfinit() ; bnf member function C T bnf bnf.clgp class group? bnf.clgp %69 [4, [4], [[7, 4, 5, 6; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1]]]?.??. Member functions, followed by relevant objects a1-a6, b2-b8, c4-c6 : coeff. of the curve. ell area : area ell bnf : big number field bnf, bnr clgp : class group bnf, bnr 11

12 2.3.2 bnfinit() tech bnfinit() flag tech flag 0 3 bnfinit() flag=1 flag=2 tech 3 tech=[c,c2,nrpid] c2=c GRH c2=12 GRH c tech c=c2=0.3 nrpid bnfcertify() bnfinit() GRH bnfinit() GRH bnf? bnfcertify(bnf) %14 = 1 12

13 2.4 gp gp ; if if() if(,, ) gp C if(cond){body;} 2.4.4? if(1, 1, 1/0) %1 = 1? if(1, 1/0, 1) *** division by zero if() for() for(,, ) 13

14 ? for(d=1,100,if(isfundamental(d), h=qfbclassno(d); if(mod(h,3)==mod(0,3),print(d, ", ", h)))) sum() sum(x=a,b,expr,x=0) X expr X = a X = b x? sum(n=1,100,1/n^2)+.0 %2 = n=1 1/n2 0.0 π 2 /6 = sum() sumalt(), sumdiv() funcname(arg1, arg2,...) = { local(v1, v2,...); } GP local() local() v1, v2 funcname() funcname() v1, v2 14

15 2.4.5 H(3, N) = L(1 3, χ D ) simple factor N = Df 2 r N Z ( 1) r N = Df 2, D Cohen [1] H(r, N) ζ(1 r) (N = 0), H(r, N) = L(1 r, χ D ) ( ) d f µ(d)χ D(d)d r 1 σ f 2r 1 d (N 0, 1(mod 4)), 0 otherwise, ζ(s) Riemann Bernoulli D χ D ( ) Legendre L(s, χ D ) χ D Dirichlet L Bernoulli µ( ) Möbius σ k ( ) k Cohen H(3, N) = s 2 (N s 2 ) s 2 (0) = 1/2, s 2 (n) = ( (n ) 2 2d 2) χ 4 (d), (n > 0). d d n s s 2 (n) GP χ d (a) kronecker(d,a) s2(n) = { if (n < 0, error("in s2(), arguments < 0")); if (n > 0, sumdiv(n, d, ((n/d)^2-2*d^2)*kronecker(-4,d)), 1/2) } 15

16 error() H(3, N) s2(n) h3(n) h3(n) = { local(s); s = s2(n) + 2 * sum(s=1, sqrt(n), s2(n-s^2)); (-1/126)*s } Pari-gp Unix Pari-gp CPU GMP Qt GNUplot Pari-gp [4] web page Configure make root make install /some/where Pari-gp $./Configure $ make all && make bench $ su - # cd /some/where/; make install 16

17 CVS 3 Pari-gp shell $ cvs -d :pserver:cvs@pari.math.u-bordeaux.fr:/home/cvs login $ cvs -z3 -d :pserver:cvs@pari.math.u-bordeaux.fr:/home/cvs co pari pari Pari-gp 3.2.gprc gp Pari-gp misc/gprc.dft /.gprc pari stack 2 gp pari stack 10M bytes

18 \\ Stack size : 10^7 Bytes. parisize = 10M \\ Biggest precomputed prime (= prevprime(10^6)) primelimit = 1M \\ Set timer on timer = C Pari Computer Aided Number Theory Introduction to Pari 3.4 GP2C GP C GP2C Bill Allombert GP C C gp GP2C GP2C Pari-gp gp2c CVS autoconf, automake, 18

19 Pari-gp CVS 3.1 gp2c URL CVS $ cvs -d :pserver:cvs@pari.math.u-bordeaux.fr:/home/cvs login $ cvs -z3 -d :pserver:cvs@pari.math.u-bordeaux.fr:/home/cvs co gp2c gp2c gp2c /some/where gp2c $./cvsinit $./configure && make $ su - # cd /some/where # make install gp2c GP2C gp2c GP C sample.gp GP 2 upto d Q( d) 3 c :int gp upto:int upto 19

20 gp2c $ cat sample.gp f(upto:int) = { local(c:int, d:int, h); c = 0; for(d=2,upto, if(isfundamental(-d), h=qfbclassno(-d);\ if(mod(h,3)==mod(0,3), c++))); c } $ gp2c -g sample.gp > sample.gp.c gp2c -g sample.gp.c /*-*- compile-command: "/usr/bin/gcc -c -o sample.gp.o -O3 -DGCC_INLINE -Wall -fno-strict-aliasing -fomit-frame-pointer -I/usr/local/include sample.gp.c && /usr/bin/gcc -o sample.gp.so -shared -O3 -DGCC_INLINE -Wall -fno-strict-aliasing -fomit-frame-pointer -Wl,-shared sample.gp.o -lc -lm -L/usr/local/lib -lgmp -L/usr/local/lib -lpari"; -*-*/ sample.gp.cc $ /usr/bin/gcc -c -o sample.gp.o -O3 \ -DGCC_INLINE -Wall -fno-strict-aliasing -fomit-frame-pointer \ -I/usr/local/include sample.gp.c && /usr/bin/gcc -o sample.gp.so \ -shared -O3 -DGCC_INLINE -Wall -fno-strict-aliasing \ 20

21 -fomit-frame-pointer -Wl,-shared sample.gp.o -lc -lm -L/usr/local/lib \ -lgmp -L/usr/local/lib -lpari $ ls sample.gp.so sample.gp.so* sample.gp.so gp f() install() gp gp sample.gp.c /* GP;install("f","G","f","./sample.gp.so"); GP;install("init_sample","v","init_sample","./sample.gp.so"); */ gp? install("f","g","f","./sample.gp.so");? install("init_sample","v","init_sample","./sample.gp.so");? f f: installed function f() installed function gp2c C gp gp2c 21

22 [1] Henri Cohen, Sums involving the values at negative integers of L- functions of quadratic characters, Math. Ann. 217 (1975), no. 3, MR 52 #3080 [2], A course in computational algebraic number theory, Springer-Verlag, Berlin, MR 94i:11105 [3], Advanced topics in computational number theory, Springer- Verlag, New York, MR [4] The PARI Group, Bordeaux, PARI/GP, Version 2.2, 2004, available from iwao@sci.toyama-u.ac.jp 22

Pari-gp 2006/7/12 2 1. Pari-gp 2. Microsoft Windows 3. 4. Pari-gp 5. 2 6. 7. Galois 8.

Pari-gp 2006/7/12 1 Pari-GP 2006 7 12 Pari-gp 2006/7/12 2 1. Pari-gp 2. Microsoft Windows 3. 4. Pari-gp 5. 2 6. 7. Galois 8. Pari-gp 2006/7/12 2-1 +ε. Pari-gp 2006/7/12 3 Pari-gp Pari C gp GP gp K. Belabas