/* Copyright (C) 2000 The PARI group. This file is part of the PARI/GP package. PARI/GP is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation. It is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY WHATSOEVER. Check the License for details. You should have received a copy of it, along with the package; see the file 'COPYING'. If not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */ /*********************************************************************/ /** **/ /** ARITHMETIC FUNCTIONS **/ /** (second part) **/ /** **/ /*********************************************************************/ #include "pari.h" #include "paripriv.h" static ulong _maxprime = 0; static ulong diffptrlen; /* Building/Rebuilding the diffptr table. The actual work is done by the * following two subroutines; the user entry point is the function * initprimes() below. initprimes1() is the old algorithm, called when * maxnum (size) is moderate. Must be called after pari_init_stack() )*/ static void initprimes1(ulong size, long *lenp, ulong *lastp, byteptr p1) { pari_sp av = avma; long k; byteptr q, r, s, p = (byteptr)stack_calloc(size+2), fin = p + size; for (r=q=p,k=1; r<=fin; ) { do { r+=k; k+=2; r+=k; } while (*++q); for (s=r; s<=fin; s+=k) *s = 1; } r = p1; *r++ = 2; *r++ = 1; /* 2 and 3 */ for (s=q=p+1; ; s=q) { do q++; while (*q); if (q > fin) break; *r++ = (unsigned char) ((q-s) << 1); } *r++ = 0; *lenp = r - p1; *lastp = ((s - p) << 1) + 1; avma = av; } /* Timing in ms (Athlon/850; reports 512K of secondary cache; looks like there is 64K of quickier cache too). arena| 30m 100m 300m 1000m 2000m <-- primelimit ================================================= 16K 1.1053 1.1407 1.2589 1.4368 1.6086 24K 1.0000 1.0625 1.1320 1.2443 1.3095 32K 1.0000 1.0469 1.0761 1.1336 1.1776 48K 1.0000 1.0000 1.0254 1.0445 1.0546 50K 1.0000 1.0000 1.0152 1.0345 1.0464 52K 1.0000 1.0000 1.0203 1.0273 1.0362 54K 1.0000 1.0000 1.0812 1.0216 1.0281 56K 1.0526 1.0000 1.0051 1.0144 1.0205 58K 1.0000 1.0000 1.0000 1.0086 1.0123 60K 0.9473 0.9844 1.0051 1.0014 1.0055 62K 1.0000 0.9844 0.9949 0.9971 0.9993 64K 1.0000 1.0000 1.0000 1.0000 1.0000 66K 1.2632 1.2187 1.2183 1.2055 1.1953 68K 1.4211 1.4844 1.4721 1.4425 1.4188 70K 1.7368 1.7188 1.7107 1.6767 1.6421 72K 1.9474 1.9531 1.9594 1.9023 1.8573 74K 2.2105 2.1875 2.1827 2.1207 2.0650 76K 2.4211 2.4219 2.4010 2.3305 2.2644 78K 2.5789 2.6250 2.6091 2.5330 2.4571 80K 2.8421 2.8125 2.8223 2.7213 2.6380 84K 3.1053 3.1875 3.1776 3.0819 2.9802 88K 3.5263 3.5312 3.5228 3.4124 3.2992 92K 3.7895 3.8438 3.8375 3.7213 3.5971 96K 4.0000 4.1093 4.1218 3.9986 3.9659 112K 4.3684 4.5781 4.5787 4.4583 4.6115 128K 4.7368 4.8750 4.9188 4.8075 4.8997 192K 5.5263 5.7188 5.8020 5.6911 5.7064 256K 6.0000 6.2187 6.3045 6.1954 6.1033 384K 6.7368 6.9531 7.0405 6.9181 6.7912 512K 7.3158 7.5156 7.6294 7.5000 7.4654 768K 9.1579 9.4531 9.6395 9.5014 9.1075 1024K 10.368 10.7497 10.9999 10.878 10.8201 1536K 12.579 13.3124 13.7660 13.747 13.4739 2048K 13.737 14.4839 15.0509 15.151 15.1282 3076K 14.789 15.5780 16.2993 16.513 16.3365 Now the same number relative to the model (1 + 0.36*sqrt(primelimit)/arena) * (arena <= 64 ? 1.05 : (arena-64)**0.38) [SLOW2_IN_ROOTS = 0.36, ALPHA = 0.38] arena| 30m 100m 300m 1000m 2000m <-- primelimit ================================================= 16K 1.014 0.9835 0.9942 0.9889 1.004 24K 0.9526 0.9758 0.9861 0.9942 0.981 32K 0.971 0.9939 0.9884 0.9849 0.9806 48K 0.9902 0.9825 0.996 0.9945 0.9885 50K 0.9917 0.9853 0.9906 0.9926 0.9907 52K 0.9932 0.9878 0.9999 0.9928 0.9903 54K 0.9945 0.9902 1.064 0.9939 0.9913 56K 1.048 0.9924 0.9925 0.993 0.9921 58K 0.9969 0.9945 0.9909 0.9932 0.9918 60K 0.9455 0.9809 0.9992 0.9915 0.9923 62K 0.9991 0.9827 0.9921 0.9924 0.9929 64K 1 1 1 1 1 66K 1.02 0.9849 0.9857 0.9772 0.9704 68K 0.8827 0.9232 0.9176 0.9025 0.8903 70K 0.9255 0.9177 0.9162 0.9029 0.8881 72K 0.9309 0.936 0.9429 0.9219 0.9052 74K 0.9715 0.9644 0.967 0.9477 0.9292 76K 0.9935 0.9975 0.9946 0.9751 0.9552 78K 0.9987 1.021 1.021 1.003 0.9819 80K 1.047 1.041 1.052 1.027 1.006 84K 1.052 1.086 1.092 1.075 1.053 88K 1.116 1.125 1.133 1.117 1.096 92K 1.132 1.156 1.167 1.155 1.134 96K 1.137 1.177 1.195 1.185 1.196 112K 1.067 1.13 1.148 1.15 1.217 128K 1.04 1.083 1.113 1.124 1.178 192K 0.9368 0.985 1.025 1.051 1.095 256K 0.8741 0.9224 0.9619 0.995 1.024 384K 0.8103 0.8533 0.8917 0.9282 0.9568 512K 0.7753 0.8135 0.8537 0.892 0.935 768K 0.8184 0.8638 0.9121 0.9586 0.9705 1024K 0.8241 0.8741 0.927 0.979 1.03 1536K 0.8505 0.9212 0.9882 1.056 1.096 2048K 0.8294 0.8954 0.9655 1.041 1.102 */ #ifndef SLOW2_IN_ROOTS /* SLOW2_IN_ROOTS below 3: some slowdown starts to be noticable * when things fit into the cache on Sparc. * The choice of 2.6 gives a slowdown of 1-2% on UltraSparcII, * but makes calculations for "maximum" of 436273009 * fit into 256K cache (still common for some architectures). * * One may change it when small caches become uncommon, but the gain * is not going to be very noticable... */ # ifdef i386 /* gcc defines this? */ # define SLOW2_IN_ROOTS 0.36 # else # define SLOW2_IN_ROOTS 2.6 # endif #endif #ifndef CACHE_ARENA # ifdef i386 /* gcc defines this? */ /* Due to smaller SLOW2_IN_ROOTS, smaller arena is OK; fit L1 cache */ # define CACHE_ARENA (63 * 1024UL) /* No slowdown even with 64K L1 cache */ # else # define CACHE_ARENA (200 * 1024UL) /* No slowdown even with 256K L2 cache */ # endif #endif #define CACHE_ALPHA (0.38) /* Cache performance model parameter */ #define CACHE_CUTOFF (0.018) /* Cache performance not smooth here */ static double slow2_in_roots = SLOW2_IN_ROOTS; typedef struct { ulong arena; double power; double cutoff; } cache_model_t; static cache_model_t cache_model = { CACHE_ARENA, CACHE_ALPHA, CACHE_CUTOFF }; /* Assume that some calculation requires a chunk of memory to be accessed often in more or less random fashion (as in sieving). Assume that the calculation can be done in steps by subdividing the chunk into smaller subchunks (arenas) and treating them separately. Assume that the overhead of subdivision is equivalent to the number of arenas. Find an optimal size of the arena taking into account the overhead of subdivision, and the overhead of arena not fitting into the cache. Assume that arenas of size slow2_in_roots slows down the calculation 2x (comparing to very big arenas; when cache hits do not matter). Since cache performance varies wildly with architecture, load, and wheather (especially with cache coloring enabled), use an idealized cache model based on benchmarks above. Assume that an independent region of FIXED_TO_CACHE bytes is accessed very often concurrently with the arena access. */ static ulong good_arena_size(ulong slow2_size, ulong total, ulong fixed_to_cache, cache_model_t *cache_model) { ulong asize, cache_arena = cache_model->arena; double Xmin, Xmax, A, B, C1, C2, D, V; double alpha = cache_model->power, cut_off = cache_model->cutoff; /* Estimated relative slowdown, with overhead = max((fixed_to_cache+arena)/cache_arena - 1, 0): 1 + slow2_size/arena due to initialization overhead; max(1, 4.63 * overhead^0.38 ) due to footprint > cache size. [The latter is hard to substantiate theoretically, but this function describes benchmarks pretty close; it does not hurt that one can minimize it explicitly too ;-). The switch between different choices of max() happens when overhead=0.018.] Thus the problem is minimizing (1 + slow2_size/arena)*overhead**0.29. This boils down to F=((X+A)/(X+B))X^alpha, X=overhead, B = (1 - fixed_to_cache/cache_arena), A = B + slow2_size/cache_arena, alpha = 0.38, and X>=0.018, X>-B. We need to find the rightmost root of (X+A)*(X+B) - alpha(A-B)X to the right of 0.018 (if such exists and is below Xmax). Then we manually check the remaining region [0, 0.018]. Since we cannot trust the purely-experimental cache-hit slowdown function, as a sanity check always prefer fitting into the cache (or "almost fitting") if F-law predicts that the larger value of the arena provides less than 10% speedup. */ /* The simplest case: we fit into cache */ if (total + fixed_to_cache <= cache_arena) return total; /* The simple case: fitting into cache doesn't slow us down more than 10% */ if (cache_arena - fixed_to_cache > 10 * slow2_size) { asize = cache_arena - fixed_to_cache; if (asize > total) asize = total; /* Automatically false... */ return asize; } /* Slowdown of not fitting into cache is significant. Try to optimize. Do not be afraid to spend some time on optimization - in trivial cases we do not reach this point; any gain we get should compensate the time spent on optimization. */ B = (1 - ((double)fixed_to_cache)/cache_arena); A = B + ((double)slow2_size)/cache_arena; C2 = A*B; C1 = (A + B - 1/alpha*(A - B))/2; D = C1*C1 - C2; if (D > 0) V = cut_off*cut_off + 2*C1*cut_off + C2; /* Value at CUT_OFF */ else V = 0; /* Peacify the warning */ Xmin = cut_off; Xmax = ((double)total - fixed_to_cache)/cache_arena; /* Two candidates */ if ( D <= 0 || (V >= 0 && C1 + cut_off >= 0) ) /* slowdown increasing */ Xmax = cut_off; /* Only one candidate */ else if (V >= 0 && /* slowdown concave down */ ((Xmax + C1) <= 0 || (Xmax*Xmax + 2*C1*Xmax + C2) <= 0)) /* DO NOTHING */; /* Keep both candidates */ else if (V <= 0 && (Xmax*Xmax + 2*C1*Xmax + C2) <= 0) /* slowdown decreasing */ Xmin = cut_off; /* Only one candidate */ else /* Now we know: 2 roots, the largest is in CUT_OFF..Xmax */ Xmax = sqrt(D) - C1; if (Xmax != Xmin) { /* Xmin == CUT_OFF; Check which one is better */ double v1 = (cut_off + A)/(cut_off + B); double v2 = 2.33 * (Xmax + A)/(Xmax + B) * pow(Xmax, alpha); if (1.1 * v2 >= v1) /* Prefer fitting into the cache if slowdown < 10% */ V = v1; else { Xmin = Xmax; V = v2; } } else if (B > 0) /* We need V */ V = 2.33 * (Xmin + A)/(Xmin + B) * pow(Xmin, alpha); if (B > 0 && 1.1 * V > A/B) /* Now Xmin is the minumum. Compare with 0 */ Xmin = 0; asize = (ulong)((1 + Xmin)*cache_arena - fixed_to_cache); if (asize > total) asize = total; /* May happen due to approximations */ return asize; } /* Use as in install(set_optimize,lLDG) \\ Through some M too? set_optimize(2,1) \\ disable dependence on limit \\ 1: how much cache usable, 2: slowdown of setup, 3: alpha, 4: cutoff \\ 2,3,4 are in units of 0.001 { time_primes_arena(ar,limit) = \\ ar = arena size in K set_optimize(1,floor(ar*1024)); default(primelimit, 200 000); \\ 100000 results in *larger* malloc()! gettime; default(primelimit, floor(limit)); if(ar >= 1, ar=floor(ar)); print("arena "ar"K => "gettime"ms"); } */ long set_optimize(long what, GEN g) { long ret = 0; switch (what) { case 1: ret = (long)cache_model.arena; break; case 2: ret = (long)(slow2_in_roots * 1000); break; case 3: ret = (long)(cache_model.power * 1000); break; case 4: ret = (long)(cache_model.cutoff * 1000); break; default: pari_err_BUG("set_optimize"); break; } if (g != NULL) { ulong val = itou(g); switch (what) { case 1: cache_model.arena = val; break; case 2: slow2_in_roots = (double)val / 1000.; break; case 3: cache_model.power = (double)val / 1000.; break; case 4: cache_model.cutoff = (double)val / 1000.; break; } } return ret; } /* s is odd; prime differences (starting from 5-3=2) start at known_primes[2], terminated by a 0 byte. Checks n odd numbers starting at 'start', setting bytes starting at data to 0 (composite) or 1 (prime) */ static void sieve_chunk(byteptr known_primes, ulong s, byteptr data, ulong n) { ulong p, cnt = n-1, start = s, delta = 1; byteptr q; memset(data, 0, n); start >>= 1; /* (start - 1)/2 */ start += n; /* Corresponds to the end */ /* data corresponds to start, q runs over primediffs */ for (q = known_primes + 1, p = 3; delta; delta = *++q, p += delta) { /* first odd number >= start > p and divisible by p = last odd number <= start + 2p - 2 and 0 (mod p) = p + last number <= start + p - 2 and 0 (mod 2p) = p + start+p-2 - (start+p-2) % 2p = start + 2(p - 1 - ((start-1)/2 + (p-1)/2) % p). */ long off = cnt - ((start+(p>>1)) % p); while (off >= 0) { data[off] = 1; off -= p; } } } /* assume maxnum <= 436273289 < 2^29 */ static void initprimes0(ulong maxnum, long *lenp, ulong *lastp, byteptr p1) { pari_sp av = avma; long alloced, psize; byteptr q, end, p, end1, plast, curdiff; ulong last, remains, curlow, rootnum, asize; ulong prime_above; byteptr p_prime_above; maxnum |= 1; /* make it odd. */ /* base case */ if (maxnum < 1ul<<17) { initprimes1(maxnum>>1, lenp, lastp, p1); return; } /* Checked to be enough up to 40e6, attained at 155893 */ rootnum = usqrt(maxnum) | 1; initprimes1(rootnum>>1, &psize, &last, p1); end1 = p1 + psize - 1; remains = (maxnum - last) >> 1; /* number of odd numbers to check */ /* we access primes array of psize too; but we access it consecutively, * thus we do not include it in fixed_to_cache */ asize = good_arena_size((ulong)(rootnum * slow2_in_roots), remains+1, 0, &cache_model) - 1; /* enough room on the stack ? */ alloced = (((byteptr)avma) <= ((byteptr)bot) + asize); if (alloced) p = (byteptr)pari_malloc(asize+1); else p = (byteptr)stack_malloc(asize+1); end = p + asize; /* the 0 sentinel goes at end. */ curlow = last + 2; /* First candidate: know primes up to last (odd). */ curdiff = end1; /* During each iteration p..end-1 represents a range of odd numbers. plast is a pointer which represents the last prime seen, it may point before p..end-1. */ plast = p - 1; p_prime_above = p1 + 2; prime_above = 3; while (remains) { /* cycle over arenas; performance not crucial */ unsigned char was_delta; if (asize > remains) { asize = remains; end = p + asize; } /* Fake the upper limit appropriate for the given arena */ while (prime_above*prime_above <= curlow + (asize << 1) && *p_prime_above) prime_above += *p_prime_above++; was_delta = *p_prime_above; *p_prime_above = 0; /* sentinel for sieve_chunk */ sieve_chunk(p1, curlow, p, asize); *p_prime_above = was_delta; /* restore */ p[asize] = 0; /* sentinel */ for (q = p; ; plast = q++) { /* q runs over addresses corresponding to primes */ while (*q) q++; /* use sentinel at end */ if (q >= end) break; *curdiff++ = (unsigned char)(q-plast) << 1; /* < 255 for q < 436273291 */ } plast -= asize; remains -= asize; curlow += (asize<<1); } last = curlow - ((p - plast) << 1); *curdiff++ = 0; /* sentinel */ *lenp = curdiff - p1; *lastp = last; if (alloced) pari_free(p); else avma = av; } ulong maxprime(void) { return diffptr ? _maxprime : 0; } void maxprime_check(ulong c) { if (_maxprime < c) pari_err_MAXPRIME(c); } /* We ensure 65302 <= maxnum <= 436273289: the LHS ensures modular function * have enough fast primes to work, the RHS ensures that p_{n+1} - p_n < 255 * (N.B. RHS would be incorrect since initprimes0 would make it odd, thereby * increasing it by 1) */ byteptr initprimes(ulong maxnum, long *lenp, ulong *lastp) { byteptr t; if (maxnum < 65537) maxnum = 65537; else if (maxnum > 436273289) maxnum = 436273289; t = (byteptr)pari_malloc((size_t) (1.09 * maxnum/log((double)maxnum)) + 146); initprimes0(maxnum, lenp, lastp, t); return (byteptr)pari_realloc(t, *lenp); } void initprimetable(ulong maxnum) { long len; ulong last; byteptr p = initprimes(maxnum, &len, &last), old = diffptr; diffptrlen = minss(diffptrlen, len); _maxprime = minss(_maxprime,last); /*Protect against ^C*/ diffptr = p; diffptrlen = len; _maxprime = last; if (old) free(old); } /* all init_primepointer_xx routines set *ptr to the corresponding place * in prime table */ /* smallest p >= a */ ulong init_primepointer_geq(ulong a, byteptr *pd) { ulong n, p; prime_table_next_p(a, pd, &p, &n); return p; } /* largest p < a */ ulong init_primepointer_lt(ulong a, byteptr *pd) { ulong n, p; prime_table_next_p(a, pd, &p, &n); PREC_PRIME_VIADIFF(p, *pd); return p; } /* largest p <= a */ ulong init_primepointer_leq(ulong a, byteptr *pd) { ulong n, p; prime_table_next_p(a, pd, &p, &n); if (p != a) PREC_PRIME_VIADIFF(p, *pd); return p; } /* smallest p > a */ ulong init_primepointer_gt(ulong a, byteptr *pd) { ulong n, p; prime_table_next_p(a, pd, &p, &n); if (p == a) NEXT_PRIME_VIADIFF(p, *pd); return p; } GEN boundfact(GEN n, ulong lim) { switch(typ(n)) { case t_INT: return Z_factor_limit(n,lim); case t_FRAC: { pari_sp av = avma; GEN a = Z_factor_limit(gel(n,1),lim); GEN b = Z_factor_limit(gel(n,2),lim); gel(b,2) = ZC_neg(gel(b,2)); return gerepilecopy(av, merge_factor_i(a,b)); } } pari_err_TYPE("boundfact",n); return NULL; /* not reached */ } /* NOT memory clean */ GEN Z_smoothen(GEN N, GEN L, GEN *pP, GEN *pe) { long i, j, l = lg(L); GEN e = new_chunk(l), P = new_chunk(l); for (i = j = 1; i < l; i++) { ulong p = (ulong)L[i]; long v = Z_lvalrem(N, p, &N); if (v) { P[j] = p; e[j] = v; j++; if (is_pm1(N)) { N = NULL; break; } } } P[0] = evaltyp(t_VECSMALL) | evallg(j); *pP = P; e[0] = evaltyp(t_VECSMALL) | evallg(j); *pe = e; return N; } /***********************************************************************/ /** **/ /** SIMPLE FACTORISATIONS **/ /** **/ /***********************************************************************/ /* Factor n and output [p,e,c] where * p, e and c are vecsmall with n = prod{p[i]^e[i]} and c[i] = p[i]^e[i] */ GEN factoru_pow(ulong n) { GEN f = cgetg(4,t_VEC); pari_sp av = avma; GEN F, P, E, p, e, c; long i, l; /* enough room to store <= 15 * [p,e,p^e] (OK if n < 2^64) */ (void)new_chunk((15 + 1)*3); F = factoru(n); P = gel(F,1); E = gel(F,2); l = lg(P); avma = av; gel(f,1) = p = cgetg(l,t_VECSMALL); gel(f,2) = e = cgetg(l,t_VECSMALL); gel(f,3) = c = cgetg(l,t_VECSMALL); for(i = 1; i < l; i++) { p[i] = P[i]; e[i] = E[i]; c[i] = upowuu(p[i], e[i]); } return f; } static GEN factorlim(GEN n, ulong lim) { return lim? Z_factor_limit(n, lim): Z_factor(n); } /* factor p^n - 1, assuming p prime. If lim != 0, limit factorization to * primes <= lim */ GEN factor_pn_1_limit(GEN p, long n, ulong lim) { pari_sp av = avma; GEN A = factorlim(subiu(p,1), lim), d = divisorsu(n); long i, pp = itos_or_0(p); for(i=2; i 1) { if (B) { gel(B,1) = shallowconcat(gel(B,1), p); gel(B,2) = shallowconcat(gel(B,2), utoipos(e-1)); } else B = to_mat(p, e-1); } } avma = av; if (!B) return v? to_mat(gen_2, v): trivial_fact(); A = cgetg(3, t_MAT); P = gel(B,1); E = gel(B,2); l = lg(P); AP = cgetg(l+1, t_COL); gel(A,1) = AP; AP++; AE = cgetg(l+1, t_COL); gel(A,2) = AE; AE++; /* prepend "2^v" */ gel(AP,0) = gen_2; gel(AE,0) = utoipos(v); for (i = 1; i < l; i++) { gel(AP,i) = icopy(gel(P,i)); gel(AE,i) = icopy(gel(E,i)); } return A; } #endif /***********************************************************************/ /** **/ /** CHECK FACTORIZATION FOR ARITHMETIC FUNCTIONS **/ /** **/ /***********************************************************************/ static int RgV_is_ZVpos(GEN v) { long i, l = lg(v); for (i = 1; i < l; i++) { GEN c = gel(v,i); if (typ(c) != t_INT || signe(c) <= 0) return 0; } return 1; } /* check whether v is a ZV with non-0 entries */ static int RgV_is_ZVnon0(GEN v) { long i, l = lg(v); for (i = 1; i < l; i++) { GEN c = gel(v,i); if (typ(c) != t_INT || !signe(c)) return 0; } return 1; } /* check whether v is a ZV with non-zero entries OR exactly [0] */ static int RgV_is_ZV0(GEN v) { long i, l = lg(v); for (i = 1; i < l; i++) { GEN c = gel(v,i); long s; if (typ(c) != t_INT) return 0; s = signe(c); if (!s) return (l == 2); } return 1; } static int is_Z_factor_i(GEN f) { return typ(f) == t_MAT && lg(f) == 3 && RgV_is_ZVpos(gel(f,2)); } int is_Z_factorpos(GEN f) { return is_Z_factor_i(f) && RgV_is_ZVpos(gel(f,1)); } int is_Z_factor(GEN f) { return is_Z_factor_i(f) && RgV_is_ZV0(gel(f,1)); } /* as is_Z_factorpos, also allow factor(0) */ int is_Z_factornon0(GEN f) { return is_Z_factor_i(f) && RgV_is_ZVnon0(gel(f,1)); } GEN clean_Z_factor(GEN f) { GEN P = gel(f,1); long n = lg(P)-1; if (n && equalim1(gel(P,1))) return mkmat2(vecslice(P,2,n), vecslice(gel(f,2),2,n)); return f; } /* n associated to a factorization of a positive integer: either N (t_INT) * a factorization matrix faN, or a t_VEC: [N, faN] */ GEN check_arith_pos(GEN n, const char *f) { switch(typ(n)) { case t_INT: if (signe(n) <= 0 ) pari_err_DOMAIN(f, "argument", "<=", gen_0, gen_0); return NULL; case t_VEC: if (lg(n) != 3 || typ(gel(n,1)) != t_INT) break; n = gel(n,2); /* fall through */ case t_MAT: if (!is_Z_factorpos(n)) break; return n; } pari_err_TYPE(f,n); return NULL; } /* n associated to a factorization of a non-0 integer */ GEN check_arith_non0(GEN n, const char *f) { switch(typ(n)) { case t_INT: if (!signe(n)) pari_err_DOMAIN(f, "argument", "=", gen_0, gen_0); return NULL; case t_VEC: if (lg(n) != 3 || typ(gel(n,1)) != t_INT) break; n = gel(n,2); /* fall through */ case t_MAT: if (!is_Z_factornon0(n)) break; return n; } pari_err_TYPE(f,n); return NULL; } /* n associated to a factorization of an integer */ GEN check_arith_all(GEN n, const char *f) { switch(typ(n)) { case t_INT: return NULL; case t_VEC: if (lg(n) != 3 || typ(gel(n,1)) != t_INT) break; n = gel(n,2); /* fall through */ case t_MAT: if (!is_Z_factor(n)) break; return n; } pari_err_TYPE(f,n); return NULL; } /***********************************************************************/ /** **/ /** MISCELLANEOUS ARITHMETIC FUNCTIONS **/ /** (ultimately depend on Z_factor()) **/ /** **/ /***********************************************************************/ /* set P,E from F. Check whether F is an integer and kill "factor" -1 */ static void set_fact_check(GEN F, GEN *pP, GEN *pE, int *isint) { GEN E, P; if (lg(F) != 3) pari_err_TYPE("divisors",F); P = gel(F,1); E = gel(F,2); RgV_check_ZV(E, "divisors"); *isint = RgV_is_ZV(P); if (*isint) { long i, l = lg(P); /* skip -1 */ if (l>1 && signe(gel(P,1)) < 0) { E++; P = vecslice(P,2,--l); } /* test for 0 */ for (i = 1; i < l; i++) if (!signe(gel(P,i)) && signe(gel(E,i))) pari_err_DOMAIN("divisors", "argument", "=", gen_0, F); } *pP = P; *pE = E; } static void set_fact(GEN F, GEN *pP, GEN *pE) { *pP = gel(F,1); *pE = gel(F,2); } int divisors_init(GEN n, GEN *pP, GEN *pE) { long i,l; GEN E, P, e; int isint; switch(typ(n)) { case t_INT: if (!signe(n)) pari_err_DOMAIN("divisors", "argument", "=", gen_0, gen_0); set_fact(absi_factor(n), &P,&E); isint = 1; break; case t_VEC: if (lg(n) != 3) pari_err_TYPE("divisors",n); set_fact_check(gel(n,2), &P,&E, &isint); break; case t_MAT: set_fact_check(n, &P,&E, &isint); break; default: set_fact(factor(n), &P,&E); isint = 0; break; } l = lg(P); e = cgetg(l, t_VECSMALL); for (i=1; i> 1)); } return mkvec2(c,f); } GEN corepartial(GEN n, long all) { pari_sp av = avma; if (typ(n) != t_INT) pari_err_TYPE("corepartial",n); return gerepileuptoint(av, corefa(Z_factor_limit(n,all))); } GEN core2partial(GEN n, long all) { pari_sp av = avma; if (typ(n) != t_INT) pari_err_TYPE("core2partial",n); return gerepilecopy(av, core2fa(Z_factor_limit(n,all))); } static GEN core2_i(GEN n) { GEN f = core(n); if (!signe(f)) return mkvec2(gen_0, gen_1); switch(typ(n)) { case t_VEC: n = gel(n,1); break; case t_MAT: n = factorback(n); break; } return mkvec2(f, sqrtint(diviiexact(n, f))); } GEN core2(GEN n) { pari_sp av = avma; return gerepilecopy(av, core2_i(n)); } GEN core0(GEN n,long flag) { return flag? core2(n): core(n); } static long _mod4(GEN c) { long r, s = signe(c); if (!s) return 0; r = mod4(c); if (s < 0) r = 4-r; return r; } GEN coredisc(GEN n) { pari_sp av = avma; GEN c = core(n); if (_mod4(c)<=1) return c; /* c = 0 or 1 mod 4 */ return gerepileuptoint(av, shifti(c,2)); } GEN coredisc2(GEN n) { pari_sp av = avma; GEN y = core2_i(n); GEN c = gel(y,1), f = gel(y,2); if (_mod4(c)<=1) return gerepilecopy(av, y); y = cgetg(3,t_VEC); gel(y,1) = shifti(c,2); gel(y,2) = gmul2n(f,-1); return gerepileupto(av, y); } GEN coredisc0(GEN n,long flag) { return flag? coredisc2(n): coredisc(n); } long omega(GEN n) { pari_sp av = avma; GEN F, P; if ((F = check_arith_non0(n,"omega"))) { long n; P = gel(F,1); n = lg(P)-1; if (n && equalim1(gel(P,1))) n--; return n; } else if (lgefint(n) == 3) { if (n[2] == 1) return 0; F = factoru(n[2]); } else F = absi_factor(n); P = gel(F,1); avma = av; return lg(P)-1; } long bigomega(GEN n) { pari_sp av = avma; GEN F, E; if ((F = check_arith_non0(n,"bigomega"))) { GEN P = gel(F,1); long n = lg(P)-1; E = gel(F,2); if (n && equalim1(gel(P,1))) E = vecslice(E,2,n); E = ZV_to_zv(E); } else if (lgefint(n) == 3) { if (n[2] == 1) return 0; F = factoru(n[2]); E = gel(F,2); } else E = ZV_to_zv(gel(absi_factor(n), 2)); avma = av; return zv_sum(E); } /* assume f = factoru(n), possibly with 0 exponents. Return phi(n) */ ulong eulerphiu_fact(GEN f) { GEN P = gel(f,1), E = gel(f,2); long i, m = 1, l = lg(P); for (i = 1; i < l; i++) { ulong p = P[i], e = E[i]; if (!e) continue; if (p == 2) { if (e > 1) m <<= e-1; } else { m *= (p-1); if (e > 1) m *= upowuu(p, e-1); } } return m; } ulong eulerphiu(ulong n) { pari_sp av = avma; GEN F; if (!n) return 2; F = factoru(n); avma = av; return eulerphiu_fact(F); } GEN eulerphi(GEN n) { pari_sp av = avma; GEN Q, F, P, E; long i, l; if ((F = check_arith_all(n,"eulerphi"))) { F = clean_Z_factor(F); n = (typ(n) == t_VEC)? gel(n,1): factorback(F); if (lgefint(n) == 3) { ulong e; F = mkmat2(ZV_to_nv(gel(F,1)), ZV_to_nv(gel(F,2))); e = eulerphiu_fact(F); avma = av; return utoipos(e); } } else if (lgefint(n) == 3) return utoipos(eulerphiu((ulong)n[2])); else F = absi_factor(n); if (!signe(n)) return gen_2; P = gel(F,1); E = gel(F,2); l = lg(P); Q = cgetg(l, t_VEC); for (i = 1; i < l; i++) { GEN p = gel(P,i), q; ulong v = itou(gel(E,i)); q = subiu(p,1); if (v != 1) q = mulii(q, v == 2? p: powiu(p, v-1)); gel(Q,i) = q; } return gerepileuptoint(av, ZV_prod(Q)); } static GEN numdiv_aux(GEN F) { GEN x, E = gel(F,2); long i, l = lg(E); x = cgetg(l, t_VECSMALL); for (i=1; i 1; v--) u = addsi(1, mului(p, u)); return u; } /* 1 + q + ... + q^v */ static GEN euler_sumdiv(GEN q, long v) { GEN u = addui(1, q); for (; v > 1; v--) u = addui(1, mulii(q, u)); return u; } static GEN u_euler_sumdivk(ulong p, long v, long k) { return euler_sumdiv(powuu(p,k), v); } static GEN euler_sumdivk(GEN p, long v, long k) { return euler_sumdiv(powiu(p,k), v); } static GEN sumdiv_aux(GEN F) { GEN x, P = gel(F,1), E = gel(F,2); long i, l = lg(P); x = cgetg(l, t_VEC); for (i=1; i 0) return gerepileuptoint(av, P); return gerepileupto(av, gdiv(P, powiu(n,k))); } /* K t_VECSMALL of k >= 0 */ GEN usumdivkvec(ulong n, GEN K) { pari_sp av = avma; GEN F = factoru(n), P = gel(F,1), E = gel(F,2), Z, S; long i,j, l = lg(P), lK = lg(K); Z = cgetg(l, t_VEC); S = cgetg(lK, t_VEC); for (j=1; j 1) return 0; return 1; } long uissquarefree(ulong n) { if (!n) return 0; return moebiusu(n)? 1: 0; } long Z_issquarefree(GEN n) { switch(lgefint(n)) { case 2: return 0; case 3: return uissquarefree(n[2]); } return moebius(n)? 1: 0; } long issquarefree(GEN x) { pari_sp av; GEN d; switch(typ(x)) { case t_INT: return Z_issquarefree(x); case t_POL: if (!signe(x)) return 0; av = avma; d = RgX_gcd(x, RgX_deriv(x)); avma = av; return (lg(d) == 3); default: pari_err_TYPE("issquarefree",x); return 0; /* not reached */ } } /*********************************************************************/ /** **/ /** DIGITS / SUM OF DIGITS **/ /** **/ /*********************************************************************/ static ulong DS[] ={ 0,1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,10,2,3,4,5,6,7,8,9,10,11,3,4,5,6,7,8, 9,10,11,12,4,5,6,7,8,9,10,11,12,13,5,6,7,8,9,10,11,12,13,14,6,7,8,9,10,11, 12,13,14,15,7,8,9,10,11,12,13,14,15,16,8,9,10,11,12,13,14,15,16,17,9,10,11, 12,13,14,15,16,17,18,1,2,3,4,5,6,7,8,9,10,2,3,4,5,6,7,8,9,10,11,3,4,5,6,7,8, 9,10,11,12,4,5,6,7,8,9,10,11,12,13,5,6,7,8,9,10,11,12,13,14,6,7,8,9,10,11, 12,13,14,15,7,8,9,10,11,12,13,14,15,16,8,9,10,11,12,13,14,15,16,17,9,10,11, 12,13,14,15,16,17,18,10,11,12,13,14,15,16,17,18,19,2,3,4,5,6,7,8,9,10,11,3, 4,5,6,7,8,9,10,11,12,4,5,6,7,8,9,10,11,12,13,5,6,7,8,9,10,11,12,13,14,6,7,8, 9,10,11,12,13,14,15,7,8,9,10,11,12,13,14,15,16,8,9,10,11,12,13,14,15,16,17, 9,10,11,12,13,14,15,16,17,18,10,11,12,13,14,15,16,17,18,19,11,12,13,14,15, 16,17,18,19,20,3,4,5,6,7,8,9,10,11,12,4,5,6,7,8,9,10,11,12,13,5,6,7,8,9,10, 11,12,13,14,6,7,8,9,10,11,12,13,14,15,7,8,9,10,11,12,13,14,15,16,8,9,10,11, 12,13,14,15,16,17,9,10,11,12,13,14,15,16,17,18,10,11,12,13,14,15,16,17,18, 19,11,12,13,14,15,16,17,18,19,20,12,13,14,15,16,17,18,19,20,21,4,5,6,7,8,9, 10,11,12,13,5,6,7,8,9,10,11,12,13,14,6,7,8,9,10,11,12,13,14,15,7,8,9,10,11, 12,13,14,15,16,8,9,10,11,12,13,14,15,16,17,9,10,11,12,13,14,15,16,17,18,10, 11,12,13,14,15,16,17,18,19,11,12,13,14,15,16,17,18,19,20,12,13,14,15,16,17, 18,19,20,21,13,14,15,16,17,18,19,20,21,22,5,6,7,8,9,10,11,12,13,14,6,7,8,9, 10,11,12,13,14,15,7,8,9,10,11,12,13,14,15,16,8,9,10,11,12,13,14,15,16,17,9, 10,11,12,13,14,15,16,17,18,10,11,12,13,14,15,16,17,18,19,11,12,13,14,15,16, 17,18,19,20,12,13,14,15,16,17,18,19,20,21,13,14,15,16,17,18,19,20,21,22,14, 15,16,17,18,19,20,21,22,23,6,7,8,9,10,11,12,13,14,15,7,8,9,10,11,12,13,14, 15,16,8,9,10,11,12,13,14,15,16,17,9,10,11,12,13,14,15,16,17,18,10,11,12,13, 14,15,16,17,18,19,11,12,13,14,15,16,17,18,19,20,12,13,14,15,16,17,18,19,20, 21,13,14,15,16,17,18,19,20,21,22,14,15,16,17,18,19,20,21,22,23,15,16,17,18, 19,20,21,22,23,24,7,8,9,10,11,12,13,14,15,16,8,9,10,11,12,13,14,15,16,17,9, 10,11,12,13,14,15,16,17,18,10,11,12,13,14,15,16,17,18,19,11,12,13,14,15,16, 17,18,19,20,12,13,14,15,16,17,18,19,20,21,13,14,15,16,17,18,19,20,21,22,14, 15,16,17,18,19,20,21,22,23,15,16,17,18,19,20,21,22,23,24,16,17,18,19,20,21, 22,23,24,25,8,9,10,11,12,13,14,15,16,17,9,10,11,12,13,14,15,16,17,18,10,11, 12,13,14,15,16,17,18,19,11,12,13,14,15,16,17,18,19,20,12,13,14,15,16,17,18, 19,20,21,13,14,15,16,17,18,19,20,21,22,14,15,16,17,18,19,20,21,22,23,15,16, 17,18,19,20,21,22,23,24,16,17,18,19,20,21,22,23,24,25,17,18,19,20,21,22,23, 24,25,26,9,10,11,12,13,14,15,16,17,18,10,11,12,13,14,15,16,17,18,19,11,12, 13,14,15,16,17,18,19,20,12,13,14,15,16,17,18,19,20,21,13,14,15,16,17,18,19, 20,21,22,14,15,16,17,18,19,20,21,22,23,15,16,17,18,19,20,21,22,23,24,16,17, 18,19,20,21,22,23,24,25,17,18,19,20,21,22,23,24,25,26,18,19,20,21,22,23,24, 25,26,27 }; ulong sumdigitsu(ulong n) { ulong s = 0; while (n) { s += DS[n % 1000]; n /= 1000; } return s; } /* res=array of 9-digits integers, return \sum_{0 <= i < l} sumdigits(res[i]) */ static ulong sumdigits_block(ulong *res, long l) { long s = sumdigitsu(*--res); while (--l > 0) s += sumdigitsu(*--res); return s; } GEN sumdigits(GEN n) { pari_sp av = avma; ulong s, *res; long l; if (typ(n) != t_INT) pari_err_TYPE("sumdigits", n); l = lgefint(n); switch(l) { case 2: return gen_0; case 3: return utoipos(sumdigitsu(n[2])); } res = convi(n, &l); if ((ulong)l < ULONG_MAX / 81) { s = sumdigits_block(res, l); avma = av; return utoipos(s); } else /* Huge. Overflows ulong */ { const long L = (long)(ULONG_MAX / 81); GEN S = gen_0; while (l > L) { S = addiu(S, sumdigits_block(res, L)); res += L; l -= L; } if (l) S = addiu(S, sumdigits_block(res, l)); return gerepileuptoint(av, S); } } static void digits_dac(GEN x, GEN B, long l, GEN* z) { GEN q,r; long m; if (l==1) { *z=x; return; } m=l>>1; q=dvmdii(x,powiu(B,m),&r); digits_dac(q,B,l-m,z); digits_dac(r,B,m,z+l-m); } static void digits_dacsmall(GEN x, ulong B, long l, ulong* z) { pari_sp av = avma; GEN q,r; long m; if (l==1) { *z=itou(x); return; } m=l>>1; q=dvmdii(x,powuu(B,m),&r); digits_dacsmall(q,B,l-m,z); digits_dacsmall(r,B,m,z+l-m); avma = av; } GEN digits(GEN x, GEN B) { pari_sp av=avma; long lz; GEN z; if (typ(x)!=t_INT) pari_err_TYPE("digits",x); if (!B) B = utoi(10); else { if (typ(B)!=t_INT) pari_err_TYPE("digits",B); if (cmpis(B,2)<0) pari_err_DOMAIN("digits","B","<",gen_2,B); } if (equalis(B,2)) {avma = av; return binaire(x); } if (!signe(x)) {avma = av; return cgetg(1,t_VEC); } if (absi_cmp(x,B)<0) {avma = av; retmkvec(absi(x)); } x = absi(x); lz = logint(x,B,NULL); if (lgefint(B)>3) { z = zerovec(lz); digits_dac(x,B,lz,(GEN*)(z+1)); return gerepilecopy(av,z); } else { ulong b = B[2]; z = zero_zv(lz); digits_dacsmall(x,b,lz,(ulong*)(z+1)); return gerepileupto(av,vecsmall_to_vec(z)); } }