-
Notifications
You must be signed in to change notification settings - Fork 3
/
charles.gp
1626 lines (1430 loc) · 46.3 KB
/
charles.gp
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
timervalue = default(timer);
default(timer, 0);
\\ ***************************************************************************************************
\\ * Working space
\\ ***************************************************************************************************
isRicePoly(P:pol, verbose:bool=1)={
if(type(P)!="t_POL", error("Must be a polynomial"));
if(poldegree(P)<2, error("Degree must be >= 2"));
for(i=0,poldegree(P),
if(type(polcoeff(P,i))!="t_INT",error("Must be an integer polynomial"))
);
if(pollead(P)!=1, return(0)); \\ Must be monic
if(polcoeff(P,1), return(0)); \\ Linear coefficient must be 0
my(c=polcoeff(P,0));
if(abs(c)>=2, return(1));
if(c==0, return(0)); \\ P(0) cannot be 0
if(subst(P,variable(P),c)==0, return(0)); \\ P(P(0)) cannot be 0
if(verbose,
print("The sequence starting with "c" and iterating");
print(variable(P)," |--> ",nice(P));
print("is a superrigid divisibility sequence.")
);
1
};
addhelp(isRicePoly, "isRicePoly(P): Checks if a polynomial P fits the conditions for Proposition 3.2 in Brian Rice, 'Primitive Prime Divisors in Polynomial Arithmetic Dynamics', Integers 7:1 (2007).");
\\ ***************************************************************************************************
\\ * Uncategorized
\\ ***************************************************************************************************
ries(x)={
if(type(x) != "t_REAL", error("Bad type in ries."));
system(Str("~/mth/ries ",x))
};
addhelp(ries, "ries(x): Feed a number to RIES.");
yafu(n)={
if(type(n) != "t_INT", error("Bad type in yafu."));
system(Str("~/mth/yafu-1.34.3/yafu 'factor(",n,")'"))
};
addhelp(yafu, "yafu(n): Factor a number with yafu.");
src(funcName:genstr)={
my(cmd=Str("egrep --color=auto --after-context=10 --before-context=1 -R '^", funcName, "'"),C,t);
C=[Str(cmd, " basemath"), Str(cmd, " kernel/gmp"), Str(cmd, " ."), Str("egrep --color=auto --after-context=5 -R '#define.*", funcName, "' .")];
for(i=1,#C,
t=Str("cd ~/mth/pari/src;", C[i]);
if (#externstr(t),
system(t);
return()
)
)
};
addhelp(src, "src(funcName): Search the PARI source for a function (or #define) called funcName and display with context.");
polygonArea(v:vec)={
if(type(v)!="t_VEC", error("Not a vector!"));
if(#v<6 || #v%2, error("Need an even number of coordinates representing at least 3 points."));
abs(sum(i=1,#v/2-1,v[2*i+2]*v[2*i-1]-v[2*i]*v[2*i+1],v[#v-1]*v[2]-v[#v]*v[1]))/2
};
addhelp(polygonArea, "polygonArea(v): Find the area of the polygon with points (v[1], v[2]), (v[3], v[4]), ..., (v[#v-1], v[#v]).");
chVec(u:vec,v:vec,umod:int=0,vmod:int=0)={
if(umod,u=apply(n->Mod(n,umod),u));
if(vmod,v=apply(n->Mod(n,vmod),v));
vector(#u*#v,i,chinese(u[(i-1)%#u+1],v[(i-1)\#u+1]))
};
addhelp(chVec, "chVec(v, u, umod, vmod): Given a vector v mod vmod and a vector u mod umod, return a vector with one representative mod lcm(umod, vmod) which is u[i] mod umod and v[j] mod vmod.");
isWardC(v:vec)={
my(aux=vector(#v),t);
if(#v==0,return(1));
aux[1]=v[1];
for(n=2,#v,
t=v[n];
fordiv(n,d,if(d<n,t/=aux[d]));
if(denominator(t)>1,
print("Not a Property C sequence at n = "n", value should be a multiple of "v[n]/t);
return(0);
);
aux[n]=t
);
aux
};
addhelp(isWardC, "isWardC(v): Does the vector v have Ward's (1939) property C? If not, return 0; if so, return the auxiliary sequence.");
plt(mn,mx,ff,flags:small=0,n:small=0)={
my(o=mn\1-1,k);
if(mx-mn < 2*n || (!n && mx-mn < 500),
my(v=vector(ceil(mx)-o,i,ff(i+o)));
ploth(x=1,2,sin(x));
ploth(x=max(mn\1+eps(),mn),min(ceil(mx)-eps(),mx),
k=x\1;v[k-o]*(k+1-x)+v[k-o+1]*(x-k), flags, n
)
,
ploth(x=max(mn\1+eps(),mn),min(ceil(mx)-eps(),mx),
k=x\1;ff(k-o)*(k+1-x)+ff(k-o+1)*(x-k), flags, n
)
)
};
addhelp(plt, "plt(mn, mx, ff, flags, n): Make a high-resolution plot of the int");
shortestPath(G, startAt)={
my(n=#G[,1],dist=vector(n,i,9e99),prev=dist,Q=2^n-1);
dist[startAt]=0;
while(Q,
my(t=vecmin(vecextract(dist,Q)),u);
if(t==9e99, break);
for(i=1,#v,if(dist[i]==t && bittest(Q,i-1), u=i; break));
Q-=1<<(u-1);
for(i=1,n,
if(!G[u,i],next);
my(alt=dist[u]+G[u,i]);
if (alt < dist[i],
dist[i]=alt;
prev[i]=u;
)
)
);
dist
};
gammainv(x:real)={
if(x<5040,return(solve(y=1,8,gamma(y)-x)));
my(L=log(x));
solve(y=L/log(L),L,lngamma(y)-L)
};
addhelp(gammainv, "gammainv(x): Inverse gamma function.");
rand(b:small)={
if(b<30,b--;return(factor(random(1<<b)+1<<b)));
while(1,
my(t=1,v=List());
forprime(p=2,127, \\ prime factors with up to 7 bits
while(!random(p),t*=p; listput(v,p))
);
if (#binary(t)>b, next);
if(#binary(t)==b,return(list2fac(v)));
for(n=8,b,
my(lambda=log(n/(n-1)),x=random(1.)/exp(-lambda),k=0);
while(1,
x-=lambda^k/k!;
if(x<0,break);
listput(v,rp(n));
t*=v[#v];
if(#binary(t)>b,next(3));
if(#binary(t)==b,return(list2fac(v)));
)
);
)
};
addhelp(rand, "rand(b): Gives a random factored n-bit integer.");
coin(v:vec)={
if(type(v)!="t_VEC", error("Must be a vector"));
for(i=1,#v,
if(type(v[i])!="t_INT" || v[i]<1,error("Must be a vector of positive integers"))
);
v=vecsort(v,,8);
my(g=vecgcd(v));
if(g>1,
print("All coins are a multiple of "g"; returning largest nonrepresentable multiple.");
return(g*coin(v/g))
);
if(#v==1,return(0));
if(#v==2,return(v[1]*v[2]-v[1]-v[2]));
if(#v==3,
my(a:int=v[1],b:int=v[2],c:int=v[3],A=gcd(b,c),B=gcd(a,c),C=gcd(a,b));
if(A>1||B>1||C>1,
\\ Johnson reduction
\\ not quite right -- G(a,a,c) != G(a,c)
error("fail");
my(a1=a/B/C,b1=b/A/C,c1=c/A/B);
return(A*B*C*(coin([a1,b1,c1])+a1+b1+c1)-a-b-c)
);
error("not quite implemented")
);
error("not implemented")
};
addhelp(coin, "coin(v): What is the largest number such that change cannot be made with zero or more coins of denomination v[1], v[2], .., v[#v]? Usually called the coin problem or the Frobenius problem.");
cons(x:real)={
my(t=precision(x),s=log(x)\log(10),v=vector(t));
x /= 10^s;
for(i=1,t,
v[i]=floor(x);
x = (x - floor(x)) * 10
);
v
};
addhelp(cons, "cons(x): Converts a constant into a sequence of decimal digits.");
\\ Output format (work-in-progress):
\\ [x1, y1, x2, y2, ..., xk, yk, c]: represents density c * x^x1 / log(x)^y1 * x^x2 / log(x)^y2 * ... * x^xk / log(x)^yk
\\ [x1, y1, x2, y2, ..., xk, yk]: same, but with unknown constant.
\\ Examples:
\\ [k] Exactly k primes
\\ [] O(1) primes
\\ [1, 1, 1] (1 + o(1)) x/log(x) primes (the maximum)
\\ [1/2, 1] Theta(sqrt(x)/log(x)) primes -- not sure about which symbol to use, maybe O is better
primesin(P:pol)={
my(t=type(P));
if (t=="t_INT",
return(if (isprime(P),
print("Contains 1 prime.");
[1]
,
print("Contains 0 primes.");
[0]
));
);
if (t=="t_FRAC" || t=="t_REAL",
print("Contains 0 primes.");
return([0])
);
if (t!="t_POL", error("bad type"));
if (poldegree(P) == 1,
my(a=polcoeff(P,1),b=polcoeff(P,0),g,t);
if (type(a) == "t_INT" && type(b) == "t_INT",
g=gcd(a,b);
if (g == 1,
t=eulerphi(a);
print("Infinitely many primes (Dirichlet), density n/log n * ",1/t);
return([1,1,1/t])
);
return(if (isprime(g) && (1 - b/g) % (a/g) == 0,
print("Contains 1 prime.");
[1]
,
print("Contains 0 primes.");
[0]
));
);
);
if (poldegree(P) > 2, error("Cannot be determined"));
};
addhelp(primesin, "primesin(P): Attempts to give the asymptotic number of primes in the polynomial P, either unconditionally or under standard conjectures. The output format is a vector, which represents:\n[x1, y1, x2, y2, ..., xk, yk, c]: density c * x^x1 / log(x)^y1 * x^x2 / log(x)^y2 * ... * x^xk / log(x)^yk\n[x1, y1, x2, y2, ..., xk, yk]: same, but with unknown leading term.");
ConjectureFEmp(a:int,b:int,c:int,lim=1e6)={
my(s=0,n=0,k);
for(n=1,lim,
k=(a*n+b)*n+c;
if(k>lim,break);
s+=isprime(k)
);
2*s/Li(sqrt(lim))
};
addhelp(ConjectureFEmp, "ConjectureFEmp(a,b,c,{lim}): Helper function for ConjectureF, given integers a, b, and c as coefficients of ax^2 + bx + c.");
ConjectureF(a:int,b:int,c:int,lim=1e6)={
my(D=b^2-4*a*c,g=gcd(a,b),f);
print("Delta = "D);
if (a <= 0 || issquare(D) || gcd(g,c) > 1, return(0));
if ((a+b)%2 ==0 & c%2 == 0, return(0));
f=factor(g)[,1];
if(a+b%2, 1, 2)/sqrt(a) * prod(i=1,#f,f[i]/(f[i]-1)) * ConjectureF_C(D,lim)
};
addhelp(ConjectureF, "ConjectureF(a,b,c): Returns a number C such that there are ~ C*sqrt(x)/log(x) primes of the form an^2 + bn + c up to x, or 0 if there are o(sqrt(x)/log(x)) or Omega(sqrt(x)) such primes. Relies on the Hardy-Littlewood Conjecture F and possibly the ERH.");
\\ prodinf(i=1,p=prime(i);1-kronecker(D,p)/(p-1))
ConjectureF_C(D,lim=1e6)={
my(cf, pr=1., l=log(lim));
if (D < -4,
print("Fung & Williams 1990");
my(f=factor(oddres(-D))[,1], c = if(delta%8 == 1, 5/2, if (delta%8 == 5, 1/2, 15/16)));
\\ Corrects for the 1 - 1/p^3 (since kronecker(D,q) == 1 about half the time)
forprime(q=3,lim,if(kronecker(D,q)==1,pr*=1-2/q/(q-1)^2));
cf = -2*eint1(2*l) - (1-2*l)/2/lim^2/l^2;
pr *= 1 - cf;
print("Correction: ", cf);
return(c * Pi^3 * sqrt(-D) / 90 / qfbclassno(D) / Lquad(D, 2) * prod(i=1,#f,1-1/f[i]^4) * pr)
);
if (D > 0,
print("Jacobson 1995");
\\ Possibly needed:
\\ QFBclassno(D) = qfbclassno(D) * if (D < 0 || norm(quadunit(D)) < 0, 1, 2)
my(f=factor(oddres(D))[,1], c = if(delta%8 == 1, 5/2, if (delta%8 == 5, 1/2, 15/16)));
forprime(q=3,lim,if(kronecker(D,q)==1,pr*=1-2/q/(q-1)^2));
\\ Corrects for the 1 - 1/p^3 (since kronecker(D,q) == 1 about half the time)
cf = -2*eint1(2*l) - (1-2*l)/2/lim^2/l^2;
pr *= 1 - cf;
print("Correction: ", cf);
return(c * Pi^4 * sqrt(D) / 180 / quadregulator(D) / qfbclassno(D) / Lquad(D, 2) * prod(i=1,#f,1-1/f[i]^4) * pr)
);
if(1,
print("Keith Conrad"); \\ http://mathoverflow.net/questions/31150/calculating-the-infinite-product-from-the-hardy-littlewood-conjecture-f
my(t);
\\ Corrects for the 1 - 2/p^3
forprime(p=3,lim,t=kronecker(D,p);pr*=(1-t/(p-1))/(1-t/p));
cf = -4*eint1(2*l) - (1-2*l)/lim^2/l^2;
pr *= 1 - cf;
print("Correction: ", cf);
return(pr/Lquad(D, 1));
);
print("Naive method");
\\ cf: the correction factor that estimates suminf(p=lim,-1/p^2) over the primes
\\ Corrects for the average of (1 - 1/p) and (1 + 1/p); not very useful.
cf=1/lim/l-eint1(l);
forprime(p=3,lim,pr*=1-kronecker(D,p)/(p-1));
print("Correction: ", cf);
pr * (1 - cf)
};
addhelp(ConjectureF_C, "ConjectureF_C(D,{lim}): Helper function for ConjectureF, finding the infinite product in the Hardy-Littlewood Conjecture F.");
hurwitz(s,x)={
my(a,res,tes,in,sig,t,m,pr,lim,nn,in2,s1,s2);
sig=real(x);
if (sig>1.5,
m=floor(sig-0.5);
return (hurwitz(s,x-m)-sum(i=1,m,(x-i)^(-s)))
);
if (sig<=0,
m=ceil(-sig+0.5);
return (hurwitz(s,x+m)+sum(i=0,m-1,(x+i)^(-s)))
);
pr=precision(1.);
sig=real(s);
t=imag(s);
default(realprecision,9);
res=s-1.;
res=if(abs(res)<0.1,-1,log(res));
lim=(pr*log(10)-real((s-.5)*res)+(1.*sig)*log(2.*Pi))/2;
lim=max(2,ceil(max(lim,abs(s*1.)/2)));
nn=ceil(sqrt((lim+sig/2-.25)^2+(t*1.)^2/4)/Pi);
default(realprecision,pr+5);
a=(x+nn+0.)^(-s);
res=sum(n=0,nn-1,(x+n)^(-s),a/2);
in=x+nn;
in2=1./(in*in);
s1=2*s-1;
s2=s*(s-1);
tes=bernreal(2*lim);
forstep (k=2*lim-2,2,-2,
tes=bernreal(k)+in2*(k*k+s1*k+s2)*tes/((k+1)*(k+2))
);
tes=in*(1+in2*s2*tes/2);
res+=tes*a/(s-1);
res=precision(res,pr);
default(realprecision,pr);
res
};
\\ (VIII.2) Complex L function, vector form. Chivec is a vector of complex
\\ values, assumed to be the values from 1 to m of a periodic function. In
\\ addition, with zero sum, such as a nontrivial character, if s=1.
\\ simple implementation, for small m.
Lsimp(chivec,s)={
my(m=#chivec);
if (s==1,
-sum(r=1,m,chivec[r]*psi(r/m))/m
,
sum(r=1,m,chivec[r]*hurwitz(s,r/m))/m^s
)
};
\\ (VIII.3) Complex L function of quadratic character, simple implementation
\\ for small |D|.
Lquad(D, s)={
my(v=vector(abs(D),i,kronecker(D,i)));
Lsimp(v,s)
};
forgap(ff)={
if(default(parisize) < 1e8,
print("Not enough memory! Resizing, please rerun command.");
allocatemem(100<<20) \\ Need ~200 MB of stack space!
);
for(i=0,13,
read(Str("gap200-chunk", if(i<10,"0",""), i));
trap(e_USER,
print("User error, ending loop...");
pg=0;
return()
,
for(j=1,#pg,
ff(pg[j])
)
);
pg=0
)
};
addhelp(forgap, "forgap(ff): Runs the command (closure) ff on each prime 2 < p < 10^12 starting a prime gap of length 200 or greater.");
forpseudo(ff)={
if(default(parisize) < 2e8 && default(parisizemax) < 2e8,
print("Not enough memory! Resizing, please rerun command.");
allocatemem(200<<20) \\ Need ~200 MB of stack space!
);
for(i=0,89,
if(default(debug), print("Reading chunk ",i+1," of 90"));
read(concat("psp/psp-chunk", i));
trap(e_USER,
print("User error, ending loop...");
pspChunk=0;
return()
,
for(j=1,#pspChunk,
my(t=ff(pspChunk[j]));
if(t,pspChunk=0; return(t))
)
);
pspChunk=0
)
};
addhelp(forpseudo, "forpseudo(ff): Runs the command (closure) ff on each 2-pseudoprime up to 2^64.");
\\ Characteristic function of A
\\ output sequence b_n = 1 if n if exist k : a_k == n
\\ otherwise b_n = 0
\\ 2nd parameter is length of output sequence
trv_char(A, n=-1)={
local(B,v);
if(n<0, n = A[#A]);
B = vector(n);
for(i=1, #A,
v=A[i];
if(type(v)=="t_INT" && v>=0 && v<n, B[v+1]=1);
);
B
};
addhelp(trv_char, "trv_char(A, {n}): Characteristic function of vector A. Gives the first n terms, or all terms that can be calculated if omitted.");
\\ Euler transform
trv_euler(A)={
my(B=vector(#A-1,n,1/n),C);
B = dirmul(vecextract(A,"2.."),B);
C = exp(Ser(concat(0,B)));
for(i=1, #A, A[i] = polcoeff(C,i-1));
A
};
addhelp(trv_euler, "trv_euler(A): Euler transform of vector A.");
ways(v, n)={
my(w=vector(n));
for(i=1,n,
for(j=1,#v,
if (i > v[j],
w[i] += w[i - v[j]];
,
if (i == v[j], w[i]++);
break
)
)
);
w
};
addhelp(ways, "ways(v, n): Given an increasing vector v of positive integers, return an n-element vector w so that w[i] is the number of ways (with ordering) to sum elements of v to get i.");
/*
time(ff,lim,sz=0)={
my(tEmpty,tNull,tFunc,timeRange,appx,gg);
sz=floor(sz)+1;
(gg=n->0);
gettime();
for(n=sz,lim+sz,gg(n));
tEmpty=gettime();
for(n=sz,lim+sz,ff(n));
tFunc=gettime();
for(n=sz,lim+sz,);
tNull=gettime();
timeRange=[tFunc - tEmpty, tFunc - tNull] / lim;
appx = (timeRange[1] + timeRange[2])/2;
if (appx < .001, print("About "round(appx*1000000)" ns"),
if (appx < 1, print("About "round(appx*1000)" µs"),
if (appx < 1000, print("About "round(appx)" ms"),
print("About "round(appx/1000)" s")
)));
precision(timeRange/1000,9) \\ Convert to seconds
};
find(m,mn,startAt)={
my(w,startAt);
if(startAt==0,
startAt = m
,
startAt = (startAt \ (m + m)) * (m + m) + m
);
forstep(n=startAt,9e999,m+m,
w=ways2(n);
if(w>mn,
print(w" "n" = "fnice(n));
return([w,n])
)
)
};
intersect(v1,v2)={
my(i=1,j=1,v=[]);
while(i<=#v1&&j<=#v2,
if(v1[i]==v2[j],
v=concat(v,v1[i]);
i++;
j++
,
if(v1[i]>v2[j],j++,i++)
)
);
v
};
addhelp(intersect, "intersect(v1,v2): Intersection of the sorted sets v1 and v2.");
doesintersect(v1,v2)={
my(i=1,j=1,v=[]);
while(i<=#v1&&j<=#v2,
if(v1[i]==v2[j],
return(1)
,
if(v1[i]>v2[j],j++,i++)
)
);
0
};
ord(a,p)={
my(f=factor(p-1),T=1,q,b);
for(i=1,#f[,1],
q=f[i,1];
b=Mod(a,p)^((p-1)/q^f[i,2]);
while(b!=1,
T*=q;
b=b^q
)
);
T
};
isperf(n)={
my(f=factor(n),spf,nd,sig);
\\print([spf,nd,sig]);
sig=prod(i=1,#f[,1],
if(f[i,2]==1,f[i,1]+1,(f[i,1]^(f[i,2]+1)-1)/(f[i,1]^f[i,2]-1))
);
if(issquare(sig,&sig),
nd=prod(i=1,#f[,1],f[i,2]+1);
spf=sum(i=1,#f[,1],f[i,1]);
sig==nd*spf
,
0
)
};
vdiff(v1,v2)={
my(i=1,j=1);
while(i<=#v1 && j<=#v2,
if (v1[i] == v2[j],
i++;
j++
,
if (v1[i] < v2[j],
print("Second vector missing "v1[i]);
i++
,
print("First vector missing "v2[j]);
j++
)
)
);
};
*/
dotproduct(a:vec, b:vec)={
if(#a != #b,
error("dotproduct: vectors are not the same size")
);
sum(i=1,#a,a[i]*b[i])
};
addhelp(dotproduct, "dotproduct(a, b): Returns the dot product of vectors a and b.");
\\ ***************************************************************************************************
\\ * Polynomials *
\\ ***************************************************************************************************
testPQ(P:pol,verbose:bool=0)={
my(disc=poldisc(P),lead=pollead(P),lin,quad);
forprime(p=2,,
if(lead%p==0, next);
my(f=lift(factor(Mod(P,p))[,1]),d=Set(apply(poldegree,f)));
if (d==[1],
lin++
, d==[2],
quad++
, d==[1,2] && disc%p==0,
\\ Ignore primes dividing the discriminant, they can factor into
\\ linear and quadratic factors.
,
if(verbose,
print("Fails at p = "p": "d" with factors "f)
);
return(0)
);
if(lin + quad > 1000, break)
);
if(verbose,
print("Appears to be polyquadratic");
);
[lin,quad]
};
isPolyquadratic(P:pol)={
if(type(P) != "t_POL", error("Not a polynomial in isPolyquadratic"));
if(!isZX(P), error("Not an integer polynomial in isPolyquadratic"));
if(!polisirreducible(P),
my(f=factor(P)[,1]);
for(i=1,#f,
if(!isPolyquadratic(f[i]), return(0))
);
return(1)
);
my(d=poldegree(P));
if (d < 3, return(1));
if (d>>valuation(d,2) != 1, return(0));
if (d < 16,
my(g=polgalois(P));
return(g == [4, 1, 1, "E(4) = 2[x]2"] || g == [8, 1, 3, "E(8)=2[x]2[x]2"])
);
if(testPQ(P,0), "probably polyquadratic", 0)
};
addhelp(isPolyquadratic, "isPolyquadratic(P): Are the roots of the integer polynomial P polyquadratic, i.e., the sum of rational multiples of square roots of integers?");
finiteOrbit(f:pol, x:mp)={
my(a=pollead(f), y=variable(f), v, d, b);
if(a < 0, error("Not implemented: negative leading coefficient"));
if(type(f) != "t_POL", error("Type error in finiteOrbit: should be polynomial"));
d=poldegree(f);
b=vecsum(abs(Vec(f)[2..d+1]))/a;
if(b < 1, b = 1);
v=List([x]);
while(1,
x=subst(f,y,x);
if(x > b, return(0));
for(i=1,#v, \\ Slow loop -- need a better data structure.
if(x==v[i], return(1));
);
listput(v, x)
)
};
addhelp(finiteOrbit, "finiteOrbit(f, x): Is the forward orbit of x (under the polynomial f) finite?");
isCriticallyFinite(f:pol)={
my(cp=nfroots(,f'), d=poldegree(f), y=variable(f));
if(#cp < d-2 && sum(i=1,#roots,valuation(f,y-cp[i])) < d-1,
error("Not implemented: polynomials with irrational critical points")
);
for(i=1,#cp,
if(!finiteOrbit(f, cp[i]), return(0))
);
1
};
addhelp(isCriticallyFinite, "isCriticallyFinite(f): ");
RgX_check_ZX(P:pol, s:genstr)={
for(i=0,poldegree(P),
if(type(polcoeff(P,i)) != "t_INT", error("Type error: must be an integer polynomial in ",s))
)
};
addhelp(RgX_check_ZX, "RgX_check_ZX(P, s): Given a polynomial P, checks if its coefficients are integers and returns an error if not. s is a string with the name of the calling function (for the error message). Mimics the library function of the same name.");
hasIntegerRoot(P:pol)={
#select(x->type(x)=="t_INT", nfroots(,P)) > 0
};
addhelp(hasIntegerRoot, "hasIntegerRoot(P): Returns 1 if the polynomial P has at least one integer root and 0 otherwise.");
Eisenstein(P:pol)={
my(d=poldegree(P),x=variable(P),g,t,times);
P/=content(P);
while(1,
g=0;
for(i=0,d-1,
g=gcd(g,polcoeff(P,i));
if(g==1,break)
);
while((t=gcd(g,polcoeff(P,d)))>1,
g/=t
);
g=factor(g)[,1];
for(i=1,#g,if(polcoeff(P,0)%g[i]^2,return([g[i],times,P])));
times++;
P=subst(P,x,x+1)
)
};
addhelp(Eisenstein, "Eisenstein(P): Given an irreducible polynomial P, searches for some k such that P(x+k) can be proven irreducible by Eisenstein's criterion using a prime q. Output is [q, k, P(x+k)].");
\\ ***************************************************************************************************
\\ * Number theory *
\\ ***************************************************************************************************
sqrtformal(n:int)={
my([d,f]=core(n,1));
if(d>1,quadgen(4*d),1)*f
};
addhelp(sqrtformal, "sqrtformal(n): Returns a number (t_QUAD or t_INT) representing the square root of n.");
factordb(n)={
my(N=n,res,pr);
if(type(n)=="t_STR",N=eval(n));
if(type(N)!="t_INT", error("Bad input"));
my(t=factor(N,2^24)); \\ Check if number is already factored (or tiny)
if(#select(k->k>2^24 || ispseudoprime(k), t[,1]) == #t[,1],
return(t)
);
\\t=alarm(1,factor(N)); \\ Pretest to remove easy composites
\\if(type(t)!="t_ERROR",return(t));
if(N<1e1000 && ispseudoprime(N),
if(default(factor_proven) && !isprime(N),
warning("Input appears to be a BPSW pseudoprime, PLEASE REPORT")
,
return(Mat([N,1]))
)
);
res = extern(concat("~/mth/fdb_factor.pl ",n));
if(!res, error("Could not find factors -- did you hit your hourly limit?"));
if(type(res)=="t_VEC",res=Mat(res));
if(type(res)!="t_MAT", error("fdb_factor.pl returned a non-matrix: ", res));
pr=prod(i=1,#res~, res[i,1]^res[i,2]);
if(N!=pr,
if(N%pr, error("factordb error on input "n));
res=concat(res,[N/pr,1])
);
for(i=1,#res~,
my(p=res[i,1]);
if(if(default(factor_proven),isprime(p),ispseudoprime(p)),
if(p>2^24 && default(factor_add_primes), addprimes(p))
,
warning("composite factor C",#Str(p)," = ",precision(p*1.,9))
)
);
res
};
addhelp(factordb, "factordb(n): Look up the factorization of n on factordb.com.");
\\ Varies as n*log(log(n))/log(n)
countsemi(n:int)={
my(B=sqrtint(n),s=-binomial(primepi(B),2));
forprime(p=2,B,
s+=primepi(n\p)
);
s
};
addhelp(countsemi, "countsemi(n): Number of semiprimes up to n. Sloane's A072000.");
\\ TODO: This generates too few even semiprimes. For example, 19% of the
\\ 20-bit semiprimes are even, but only 16% of the numbers generated by rsp(20)
\\ are even:
\\ shouldbe(b)=(primepi(2^(b-1)-1)-primepi(2^(b-2)-1))/(countsemi(2^b-1)-countsemi(2^(b-1)-1))
\\ is(b)=my(M=.261497212847642783755);(log(log(3))+M)/(log(b*log(2)/2)+M)
\\
\\ This is because the first prime 2 takes essentially all the error in the
\\ approximation log log x + M on itself.
\\
\\ Other issues include: (1) primes with big prime gaps following take too much
\\ weight, (2) primes around the square root may be over- or under-represented,
\\ and (3) semiprimes are larger than 2^b on occasion. (Also under 2^(b-1), but
\\ this is much less common and not really a concern.) Problem #1 can be
\\ handled with the problem with 2 by handling small primes separately -- maybe
\\ through 127. This might require handling the semiprimes under 15 bits as
\\ special cases.
rsp(b:small,flag:small=0)={
if (bitand(flag,2),
if(b<3, error("No semiprimes with fewer than 3 bits"));
my(p,q,lower,upper,t);
if(b%2,
lower=sqrtint(1<<(b-2))+1;
upper=1<<((b+1)/2);
,
lower=1<<((b-2)/2);
upper=sqrtint(1<<(b+1));
);
while(1,
p = precprime(random(upper - lower) + lower);
q = nextprime(random(upper - lower) + lower);
if (p > q,
t = p;
p = q;
q = t;
);
if (2*p > q && #binary(p * q) == b, break)
);
return(if(bitand(flag,1),
if(p==q,
matrix(1,2,i,j,if(j==1,p,2))
,
[p,1;q,1]
)
,
p*q
))
);
if(b<26, \\ Cases below 15 must be handled here so that primes up to
\\ 127 can be handled separately. Cases below 26 are just
\\ faster this way, depending on precision.
my(n);
return(if(b<8,
if(b<3, error("No semiprimes with fewer than 3 bits"));
if(b==3,n=[4,6]);
if(b==4,n=[9,10,14,15]);
if(b==5,n=[21,22,25,26]);
if(b==6,n=[33,34,35,38,39,46,49,51,55,57,58,62]);
if(b==7,n=[65,69,74,77,82,85,86,87,91,93,94,95,106,111,115,118,119,121,122,123]);
n=n[random(#n)+1];
if(flag,factor(n),n)
,
b=2^(b-1);
while(bigomega(n=random(b)+b)!=2,); \\ Brute force
n
))
);
\\ Choose the small prime as 2 with probability 1/2, 3 with probability 1/3, etc.
my(M=.261497212847642783755,weight=log(b*log(2)/2)+M,r=random(round(weight << 64))*1.>>64,p=precprime(exp(exp(r-M))),q);
q=2^(b-1)\p;
q=nextprime(q+random(q));
if(p > q,
b = p;
p = q;
q = b;
);
if(bitand(flag,1),
if(p==q,
matrix(1,2,i,j,if(j==1,p,2))
,
[p,1;q,1]
)
,
p*q
)
};
addhelp(rsp, "rsp(b, {flag=0}): Generates a random b-bit semiprime. The bits of flag mean: 1: returns the factorization of the number instead of the number, 2: return worst-case semiprimes rather than average semiprimes.");
/*
bestappr2(x, n)={
my(b=bestappr(x, n), cf=contfrac(x, #contfrac(b)+1), lower=ceil(cf[#cf]/2), upper=cf[#cf]);
cf[#cf] = lower;
contfracback(cf)
};
addhelp(bestappr2, "bestappr2(x, n): Work-in-progress. Should determine the best rational approximation of x with denominator up to n.");
*/
\\ ***************************************************************************************************
\\ * Operations on generic lists *
\\ ***************************************************************************************************
ternarySearch(f, left, right)={
my(e=2*eps());\\left+right));
left *= 1.; \\ convert to t_REAL
while(right-left>e,
my(L=(2*left+right)/3,R=(left+2*right)/3);
if(f(L) < f(R), left=L, right=R)
);
(left+right)/2
};
addhelp(ternarySearch, "ternarySearch(f, left, right): Assuming f is unimodal, find left <= x <= right such that f(x) is maximal.");
chop(v:vec,n:int)={
if(#v<n,v,vector(n,i,v[i]))
\\ vecextract?
};
addhelp(chop, "chop(v, n): Returns the first n terms of v, or all the terms if n > #v.");
Aitken(v:vec)={
vector(#v-2, i, (v[i+2]*v[i]-v[i+1]^2)/(v[i+2]-2*v[i+1]+v[i]))
};
addhelp(Aitken, "Aitken(v): Applies Aitken's delta-squared process to v. The precision of v needs to be at least twice the desired precision to get meaningful results.");
LIP(X=0,Y)={
my(n=#Y);
if(type(X)!="t_VEC",X=vector(n,i,i)); \\ Default to X = [1,2,3,...]
if (#X!=n, error("Vectors must be the same size!"));
sum(j=1,n,
Y[j]*prod(k=1,n,if(j==k,1,(x-X[k])/(X[j]-X[k])))
)
};
addhelp(LIP, "LIP(X,Y): Gives the Lagrange interpolating polynomial for X=[x1,x2,...] and Y=[y1,y2,...]. If X is omitted, use [1,2,...].");
isperiodic(v:vec)={
\\ k is the period
for(k=1,(#v+1)\2,
for(i=k+1,#v,if(v[i]!=v[i-k], next(2)));
return(k)
);
print("Not periodic with period <= ",(#v+1)\2);
0
};
isrec(v:vec,P:pol)={
my(d=poldegree(P),c=vector(d,i,-polcoeff(P,d-i)));
if (#v < 2*d,
warning(Str("isrec: Not enough information to determine if the sequence is a ",d,"-coefficient recurrence relation; matrix is underdetermined. Give more terms or reduce d and try again."));
return
);
for(n=d+1,#v,
if(v[n] != sum(i=1,d,v[n-i] * c[i]),return(0))
);
1
};
addhelp(isrec, "isrec(v, P): Is the vector v a recurrence relation with characteristic polynomial P?");
minrec(v:vec,P:pol)={
if (!isrec(v,P),error("Sequence is not generated by the given recurrence."));
my(f=factor(P));
for(i=1,#f[,1],
for(j=1,f[i,2],
if(!isrec(v,P/f[i,1]),break);
P/=f[i,1]
)
);
P
};
addhelp(minrec, "minrec(v, P): Given a vector v which is a recurrence relation with characteristic polynomial P, return the lowest-degree polynomial which represents a recurrence relation for v.");
ggf(v)={
my( p, q, B=#v\2 ); if( B<4, "Need at least 8 values!",
if( !#q=qflll(matrix(B,B,x,y,v[x-y+B+1]),4)[1], "QFLLL returned empty result.",
polcoeff(p=Ser(q[,1]),0)<0 && p=-p; /* adjust sign */
q=Pol(Ser(v)*p);
if( Ser(v)==q/=Pol(p), q,
Str("I guessed ",q,", but it doesn't reproduce the last value!")
)))
};
pgf(f)={
my(p=P->concat( vecextract( Vec(Str( P+O(x^99) )), "..-11")));
Str("(",p(numerator(f)),")/(",p(denominator(f)),")") /* prettyprint rational g.f. */
};
\\ Helper function.
tidy(x)={
if(type(x) == "t_COMPLEX",
if(abs(imag(x))<=eps(), x=real(x));
if(abs(real(x))<=eps(), x=imag(x)*I);
return(x)
);
if(type(x) == "t_VEC",
vector(#x,i,tidy(x[i]))
,
x
)
};
\\ Only works in the case of distinct roots -- doesn't know about polynomials
findBinet(v:vec,offset:int=1)={
my(c=findrec(v,0),poly=Pol(concat(-1,c)),t=polroots(poly),roots=tidy(t),coeff);
coeff=matsolve(matrix(#c,#c,i,j,roots[i]^(j+offset-1)),vectorv(#c,i,v[i]));
coeff=tidy(coeff);