*h0A2SrSSKrSSKJr SSKJr SSKJrJrJrJrJ r J r J r S/S-r \ "SS5Hr\/SS \- --\ S\-SS\S--2'M S=S jrS rS r\S :XaSSKr\R r\R"rSr\S:Xa\R("5S:aSrOSr\ "S5Vs/sHnSU-PM snrSrSrSr\S:Xa\r\rO\S :Xa\R4r\r\rO\r\r\S:XaS\"\5;a \R<r\ "S5Vs/sH n\"U5PM snr\ "S5Vs/sH n\"U5PM snr Sr!Sr"S\"4Sjr#SS\"4Sjr$SS\"4Sjr%\S:Xa\%r&O\$r&SS-r'SS -r(SS!-r)SS"-r*S#r+S$r,S%r-S&r.S'r/S(r0S)r1\1r2\S:XaO\R("5S:a\Rf=r4=r5r3\Rlr7O>\Rp=r4=r5r3\Rnr7O!\S :Xa\9"\S*S+5=r4=r5r3S,r7O\-r4\.r5\0r3\/r704S-jr:S.r;SSS/.4S0jrzw Utility functions for integer math. TODO: rename, cleanup, perhaps move the gmpy wrapper code here from settings.py N)bisect)xrange)BACKENDgmpysage sage_utilsMPZMPZ_ONEMPZ_ZEROcdU/nUSX-:aX3SU-S-/-nUSX-:aMUSSS2$)a Return a list of integers ~= [start, n*start, ..., target/n^2, target/n, target] but conservatively rounded so that the quotient between two successive elements is actually slightly less than n. With n = 2, this describes suitable precision steps for a quadratically convergent algorithm such as Newton's method; with n = 3 steps for cubic convergence (Halley's method), etc. >>> giant_steps(50,1000) [66, 128, 253, 502, 1000] >>> giant_steps(50,1000,4) [65, 252, 1000] N)starttargetnLs T/home/mande/repo/quber/.venv/lib/python3.13/site-packages/mpmath/libmp/libintmath.py giant_stepsrsJ& A B%%'/ 2A  B%%'/ TrT7Nc US:aX- $X*-$)zFor an integer x, calculate x >> n with the fastest (floor) rounding. Unlike the plain Python expression (x >> n), n is allowed to be negative, in which case a left shift is performed.rrxrs rrshiftr+ Avaf}Birc US:aX-$X*- $)zFor an integer x, calculate x << n. Unlike the plain Python expression (x << n), n is allowed to be negative, in which case a right shift with default (floor) rounding is performed.rrrs rlshiftr!2rrrcU(dgUS-nU(a [U$SnUS-nUS-(dUS-nUS- nUS-(dMU[US--$)z1Count the number of trailing zero bits in abs(n).rr)small_trailing)rlow_bytets rpython_trailingr'>sg 4xHh'' A!GA$h a Q$hh ~a$h' ''rr2cDU(a[U5R5$gzr=ss1v||~rc4[U5R5$r?)r sqrtremr,s rrr>sA(rc:US:aSU*S--[U*5-$X;aX$Un[[[[4up4pVU(aIUS-(aX6-nXF-U-X5--XE-U-pCUS-nOXf-nXU-U-USU-U--peUS-nU(aMIUS:aXAU'U$)z?Computes the nth Fibonacci number as an integer, for integer n.rrrrrX)ifibr r ) r_cachemabrzqaqqqs rrrFs 1uqbd|dA2h&&{y A(Hg5JA! q5B3r6!#:qs2vq FAB3r62ac!e8q !GA ! 3wq Hri)rrcURU5nU(aU$[U5nXS- n[nX0::aXC-nX5::aXAU'US- nX0::aMU$)z.Return n factorial (for integers n >= 0 only).r)getlenMAX_FACTORIAL_CACHE)rmemofkrzMAXs rifacras_  A D A qS A C &  8G Q & HrcXS-nURU5nU(aU$[U5nX$n[nX@:aUS- nXT-nXF::aXRU'X@:aMU$)z4Return n!! (double factorial), integers n >= 0 only.rr)rmaxr)r memo_pairrrrrzrs rifac2rpsf qS>D  A D A A C % Q  8G % Hrc@[[R"U55$r?)r5r factorialr,s rrrsS*+rcUS-n[[U55nSS/USS&[S[US-5S-5H(nX(dM[US-X5HnSX'M M* UVs/sH oD(dM UPM sn$s snf)Nrrrrg)listrr5)rsieveijrzs r list_primesrs AA OEAE"1I As1c6{1} % 88AqD!'(& "u!Au "" "s , B:Bcr[R"US-5Vs/sHn[U5PM sn$s snf)Nr)rprimesr5)r_s rrrs- $ AaC 01 01A 0111s4)rqr %)+/c4^^^^[T5mTS-(dTS:H$TS:a T[;$[HnTU-(aM g TS- m[T5mTT- mUUUU4SjnTS:aSS/nOTS:a/S QnO[nUHnU"U5(aM g g ) a Determines whether n is a prime number. A probabilistic test is performed if n is very large. No special trick is used for detecting perfect powers. >>> sum(list_primes(100000)) 454396537 >>> sum(n*isprime(n) for n in range(100000)) 454396537 rrriFc>[UTT5nUS:XdUT:Xag[ST5HnUS-T-nUT:XdM g g)NrTrF)powr)rrrmdrrss rtestisprime..testsL !AJ 6Q!V!A1qAAvrirqlHe%Z )rrqrrrrrT)r5small_odd_primes_setsmall_odd_primestrailing)rrzr witnessesrrrrs` @@@risprimers AA q5Av 2v((( 1uu !A A QA 7{qE _ & $ Aww rc^[[U55nUS:aU$/n[SUS-5HKmUT-(aMUTS--(d g[U4SjU55(aM:UR T5 MM S[ U5-$)z Evaluates the Moebius function which is `mu(n) = (-1)^k` if `n` is a product of `k` distinct primes and `mu(n) = 0` otherwise. TODO: speed up using factorization rrrc3.># UH nTU-v M g7fr?r).0rrzs r moebius..s.gq1ugsr)absr5rsumrNr)rfactorsrzs @rmoebiusrs| CF A1uG Aqs^A1H.g...q!  W rc`SnUH%nU(aU(aX!U-p!U(aMM!M#UnM' U$)Nrr)argsrrs rgcdrs5 A  a%1!A  HricUS-(a[$URU5nU(aU$[nUnSVs/sHn[U5PM nn[ SUS-5Hn[ US-SS5H nUS- Xg-US-XgS---XgS-'M" UR S5 Sn[ US-SS5H'n XU S-- nXC::dMSUS--USU---X'M) X@:XdMSUS--U-SU--s $ gs snf)a Computes the Euler numbers `E(n)`, which can be defined as coefficients of the Taylor expansion of `1/cosh x`: .. math :: \frac{1}{\cosh x} = \sum_{n=0}^\infty \frac{E_n}{n!} x^n Example:: >>> [int(eulernum(n)) for n in range(11)] [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521] >>> [int(eulernum(n)) for n in range(11)] # test cache [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521] r)rrrrrrrrrN)r rMAX_EULER_CACHEr rangerN) rrrrrrrrsumars reulernumrs$ 1u 1 A C A&'AQA'AqsmqsB#Ac14Z1Q3A#,.AcF$  qsB#A acFNDx AqD\DAqDL9 $ 61a4L$&!Q$. . (sC?c4US:dUS:a[eX:a [X:H5$US:a[$[/US--n[US'[ SUS-5H4n[ [ X5SS5HnUS- X$-X$S- -X$'M M6 SX--X!-$)z$ Stirling number of the first kind. rrrr) ValueErrorr r r rru)rrrrrs r stirling1r%s 1uAv16{1u  acA AaD Aqs^A 1b)AaC14rs[ LLLs qA&'SA!H%5N1a4>QqS>" 0   f __F __F ( f ||~    c # 1!Q$ #' fHH ''MHHHH fT2H$): .:ahqk: . %d , 18A; ,- 3 Y Ai,!I, fGG C C C C  4.*`(   f ||~+/::5 5j5..+/994 4j5,,  G%=>?K?*u(G$K"J EG 2  1!u~  f 88D  +D >>D# f2<+,*X& J'{$/L"{ $J/ ,s2J J:J