क्रमगुणित

गणितमें किसी अऋणात्मकपूर्णांकnकाक्रमगुणितया 'फैक्टोरियल' वह संख्या है जो उस पूर्णांकnतथा उससे छोटे सभी धनात्मक पूर्णांकों के गुननफल के बराबर होता है। इसेn!, से निरूपित किया जाता है। उदाहरण के लिये,
0!का मान is1होता है।
गणित के अनेकों क्षेत्रों में क्रमगुणित का उपयोग करना पड़ता है, जिनमें सेक्रमचय-संचय,बीजगणिततथा गणितीय विश्लेषण प्रमुख हैं।
क्रमगुणित की अवधारणा कई संस्कृतियों में स्वतंत्र रूप से उत्पन्न हुई है:
15वीं शताब्दी के उत्तरार्ध से क्रमगुणित पश्चिमी गणितज्ञों के अध्ययन का विषय बन गया। 1494 के एक ग्रंथ में, इतालवी गणितज्ञ लुका पसिओली ने डाइनिंग टेबल व्यवस्था की एक समस्या के संबंध में 11! तक के क्रमगुणित की गणना की।क्रिस्टोफर क्लैवियसनेजोहान्स डी सैक्रोबोस्कोके काम पर 1603 की एक टिप्पणी में फैक्टोरियल पर चर्चा की, और 1640 के दशक में, फ्रांसीसी पॉलीमैथमैरिन मर्सेनने क्लैवियस के काम के आधार पर, 64! तक के फैक्टोरियल की बड़ी (लेकिन पूरी तरह से सही नहीं) सारणियां प्रकाशित कीं।घातांकीय फलनके लिएघात श्रेणी, इसके गुणांकों के लिए फैक्टोरियल के पारस्परिक के साथ, पहली बार 1676 मेंआइजैक न्यूटनद्वारागॉटफ्रीड विल्हेम लीबनिजको लिखे एक पत्र में तैयार किया गया था।क्रमगुणित पर प्रारंभिक यूरोपीय गणित के अन्य महत्वपूर्ण कार्यों मेंजॉन वालिसद्वारा 1685 के ग्रंथ में व्यापक कवरेज,अब्राहम डी मोइवरद्वारा 1721 मेंn{\displaystyle n}के बड़े मूल्यों के लिए उनके अनुमानित मूल्यों का एक अध्ययन,जेम्स स्टर्लिंगद्वारा डी मोइवर को 1729 का एक पत्र जिसमें बताया गया था किस्टर्लिंग का सन्निकटनके रूप में क्या जाना जाता है, औरडैनियल बर्नौलीऔरलियोनहार्ड यूलरद्वारा उसी समय क्रमगुणित फलन केगामा फलनतक निरंतर विस्तार को सूत्रबद्ध करने का कार्य शामिल है।एड्रियन-मैरी लीजेंड्रेनेलीजेंड्रे का सूत्रशामिल किया, जिसमेंअभाज्य घातोंमें क्रमगुणित केकारकीकरणमें घातांकों का वर्णन किया गया था,संख्या सिद्धांतपर 1808 के एक पाठ में।
क्रमगुणित के लिए संकेतनn!{\displaystyle n!}फ्रांसीसी गणितज्ञक्रिश्चियन क्रैम्पद्वारा 1808 में पेश किया गया था।कई अन्य संकेतन भी इस्तेमाल किए गए हैं। बाद के एक और संकेतन|n_{\displaystyle \vert \!{\underline {\,n}}}, जिसमें क्रमगुणित का तर्क एक बॉक्स के बाएं और निचले हिस्से से आधा घिरा हुआ था, ब्रिटेन और अमेरिका में कुछ समय के लिए लोकप्रिय था, लेकिन इसका इस्तेमाल बंद हो गया, शायद इसलिए क्योंकि इसे टाइप करना मुश्किल है।"फैक्टोरियल" शब्द (मूल रूप से फ्रेंच:फैक्टोरियल) का पहली बार प्रयोग 1800 मेंलुई फ्रांकोइस एंटोनी आर्बोगास्टद्वारा किया गया था,फाआ डी ब्रूनो के सूत्रपर पहले काम में,लेकिनअंकगणितीय प्रगतिके उत्पादों की एक अधिक सामान्य अवधारणा का उल्लेख करते हुए। यह नाम जिन "कारकों" को संदर्भित करता है, वे क्रमगुणित के लिए गुणन सूत्र के पद हैं।
एक धनात्मक पूर्णांकn{\displaystyle n}का क्रमगुणित फलनn{\displaystyle n}से छोटे सभी धनात्मक पूर्णांकों के गुणनफल द्वारा परिभाषित किया जाता है।n!=1⋅2⋅3⋯(n−2)⋅(n−1)⋅n.{\displaystyle n!=1\cdot 2\cdot 3\cdots (n-2)\cdot (n-1)\cdot n.}इसेउत्पाद संकेतनमें और अधिक संक्षिप्त रूप से इस प्रकार लिखा जा सकता हैn!=∏i=1ni.{\displaystyle n!=\prod _{i=1}^{n}i.}
यदि इस गुणन सूत्र को अंतिम पद को छोड़कर बाकी सभी पदों को रखने के लिए बदल दिया जाता है, तो यह एक छोटे क्रमगुणित के लिए समान रूप का एक गुणनफल परिभाषित करेगा। यह एकपुनरावृत्ति संबंधकी ओर ले जाता है, जिसके अनुसार क्रमगुणित फलन का प्रत्येक मान पिछले मान कोn{\displaystyle n}से गुणा करके प्राप्त किया जा सकता है:n!=n⋅(n−1)!.{\displaystyle n!=n\cdot (n-1)!.}उदाहरण के लिए,5!=5⋅4!=5⋅24=120{\displaystyle 5!=5\cdot 4!=5\cdot 24=120}
0{\displaystyle 0}काक्रमगुणित1{\displaystyle 1}है,या प्रतीकों में,0!=1{\displaystyle 0!=1}है।इस परिभाषा के लिए कई प्रेरणाएँ हैं:
The earliest uses of the factorial function involve countingpermutations: there aren!{\displaystyle n!}different ways of arrangingn{\displaystyle n}distinct objects into a sequence.Factorials appear more broadly in many formulas incombinatorics, to account for different orderings of objects. For instance thebinomial coefficients(nk){\displaystyle {\tbinom {n}{k}}}count thek{\displaystyle k}-elementcombinations(subsets ofk{\displaystyle k}elements)from a set withn{\displaystyle n}elements,and can be computed from factorials using the formula(nk)=n!k!(n−k)!.{\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}.}TheStirling numbers of the first kindsum to the factorials, and count the permutationsofn{\displaystyle n}grouped into subsets with the same numbers of cycles.Another combinatorial application is in countingderangements, permutations that do not leave any element in its original position; the number of derangements ofn{\displaystyle n}items is thenearest integerton!/e{\displaystyle n!/e}.
Inalgebra, the factorials arise through thebinomial theorem, which uses binomial coefficients to expand powers of sums.They also occur in the coefficients used to relate certain families of polynomials to each other, for instance inNewton's identitiesforsymmetric polynomials.Their use in counting permutations can also be restated algebraically: the factorials are theordersof finitesymmetric groups.Incalculus, factorials occur inFaà di Bruno's formulafor chaining higher derivatives.Inmathematical analysis, factorials frequently appear in the denominators ofpower series, most notably in the series for theexponential function,ex=1+x1+x22+x36+⋯=∑i=0∞xii!,{\displaystyle e^{x}=1+{\frac {x}{1}}+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+\cdots =\sum _{i=0}^{\infty }{\frac {x^{i}}{i!}},}and in the coefficients of otherTaylor series(in particular those of thetrigonometricandhyperbolic functions), where they cancel factors ofn!{\displaystyle n!}coming from then{\displaystyle n}th derivativeofxn{\displaystyle x^{n}}.This usage of factorials in power series connects back toanalytic combinatoricsthrough theexponential generating function, which for acombinatorial classwithni{\displaystyle n_{i}}elements ofsizei{\displaystyle i}is defined as the power series∑i=0∞xinii!.{\displaystyle \sum _{i=0}^{\infty }{\frac {x^{i}n_{i}}{i!}}.}
Innumber theory, the most salient property of factorials is thedivisibilityofn!{\displaystyle n!}by all positive integers upton{\displaystyle n},described more precisely for prime factors byLegendre's formula. It follows that arbitrarily largeprime numberscan be found as the prime factors of the numbersn!±1{\displaystyle n!\pm 1}, leading to a proof ofEuclid's theoremthat the number of primes is infinite.Whenn!±1{\displaystyle n!\pm 1}is itself prime it is called afactorial prime;relatedly,Brocard's problem, also posed bySrinivasa Ramanujan, concerns the existence ofsquare numbersof the formn!+1{\displaystyle n!+1}.In contrast, the numbersn!+2,n!+3,…n!+n{\displaystyle n!+2,n!+3,\dots n!+n}must all be composite, proving the existence of arbitrarily largeprime gaps.An elementaryproof of Bertrand's postulateon the existence of a prime in any interval of theform[n,2n]{\displaystyle [n,2n]},one of the first results ofPaul Erdős, was based on the divisibility properties of factorials.Thefactorial number systemis amixed radixnotation for numbers in which the place values of each digit are factorials.
Factorials are used extensively inprobability theory, for instance in thePoisson distributionand in the probabilities ofrandom permutations.Incomputer science, beyond appearing in the analysis ofbrute-force searchesover permutations,factorials arise in thelower boundoflog2⁡n!=nlog2⁡n−O(n){\displaystyle \log _{2}n!=n\log _{2}n-O(n)}on the number of comparisons needed tocomparison sorta set ofn{\displaystyle n}items,and in the analysis of chainedhash tables, where the distribution of keys per cell can be accurately approximated by a Poisson distribution.Moreover, factorials naturally appear in formulae fromquantumandstatistical physics, where one often considers all the possible permutations of a set of particles. Instatistical mechanics, calculations ofentropysuch asBoltzmann's entropy formulaor theSackur–Tetrode equationmust correct the count ofmicrostatesby dividing by the factorials of the numbers of each type ofindistinguishable particleto avoid theGibbs paradox. Quantum physics provides the underlying reason for why these corrections are necessary.
n{\displaystyle n}के एक फलन के रूप में, क्रमगुणितचरघातांकीय वृद्धिकी तुलना में तेज़ है, लेकिनदोहरे चरघातांकीय फलनकी तुलना में अधिक धीरे-धीरे बढ़ता है।इसकी वृद्धि दरnn{\displaystyle n^{n}}के समान है, लेकिन एक घातांकीय कारक द्वारा धीमी है। इस परिणाम तक पहुँचने का एक तरीका क्रमगुणित काप्राकृतिक लघुगणकलेना है, जो इसके उत्पाद सूत्र को योग में बदल देता है, और फिर एक समाकल द्वारा योग का अनुमान लगाता है:ln⁡n!=∑x=1nln⁡x≈∫1nln⁡xdx=nln⁡n−n+1{\displaystyle \ln n!=\sum _{x=1}^{n}\ln x\approx \int _{1}^{n}\ln x\,dx=n\ln n-n+1}परिणाम को घातांकित करना (और नगण्य+1{\displaystyle +1}पद को अनदेखा करना)n!{\displaystyle n!}को(n/e)n{\displaystyle (n/e)^{n}}के रूप में सन्निकट करता है।}}ट्रेपेज़ॉइड नियमका उपयोग करके, ऊपर और नीचे दोनों योग को एक समाकल द्वारा अधिक सावधानी से बांधने से पता चलता है कि इस अनुमान को एक सुधार कारक की आवश्यकता है जोn{\displaystyle {\sqrt {n}}}के समानुपाती हो। इस सुधार के लिए आनुपातिकता का स्थिरांकवालिस उत्पादसे पाया जा सकता है, जोπ{\displaystyle \pi }को क्रमगुणित और दो की घातों के सीमित अनुपात के रूप में व्यक्त करता है। इन सुधारों का परिणामस्टर्लिंग का सन्निकटनहै:n!∼2πn(ne)n{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\,}यहाँ,∼{\displaystyle \sim }प्रतीक का अर्थ है कि, जैसे-जैसेn{\displaystyle n}अनंत की ओर जाता है, बाएँ और दाएँ पक्षों के बीच का अनुपातसीमामें एक के करीब पहुँचता है।
स्टर्लिंग का सूत्रअसममितीय श्रृंखलामें पहला पद प्रदान करता है जो अधिक संख्या में पदों के लिए ले जाने पर और भी अधिक सटीक हो जाता है:n!∼2πn(ne)n(1+112n+1288n2−13951840n3−5712488320n4+⋯){\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+{\frac {1}{12n}}+{\frac {1}{288n^{2}}}-{\frac {139}{51840n^{3}}}-{\frac {571}{2488320n^{4}}}+\cdots \right)}एक वैकल्पिक संस्करण सुधार पदों में केवल विषम घातांक का उपयोग करता है:n!∼2πn(ne)nexp⁡(112n−1360n3+11260n5−11680n7+⋯){\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\exp \left({\frac {1}{12n}}-{\frac {1}{360n^{3}}}+{\frac {1}{1260n^{5}}}-{\frac {1}{1680n^{7}}}+\cdots \right)}इन सूत्रों के कई अन्य रूप भीश्रीनिवास रामानुजन,बिल गोस्पर, और अन्य द्वारा विकसित किए गए हैं।
तुलना सॉर्टिंगका विश्लेषण करने के लिए उपयोग किए जाने वाले क्रमगुणित केबाइनरी लघुगणकको स्टर्लिंग के सन्निकटन का उपयोग करके बहुत सटीक रूप से अनुमानित किया जा सकता है। नीचे दिए गए सूत्र में,O(1){\displaystyle O(1)}पदबड़ा O संकेतनको आमंत्रित करता है।log2⁡n!=nlog2⁡n−(log2⁡e)n+12log2⁡n+O(1){\displaystyle \log _{2}n!=n\log _{2}n-(\log _{2}e)n+{\frac {1}{2}}\log _{2}n+O(1)}
The product formula for the factorial implies thatn!{\displaystyle n!}isdivisibleby allprime numbersthat are atmostn{\displaystyle n},and by no larger prime numbers.More precise information about its divisibility is given byLegendre's formula, which gives the exponent of each primep{\displaystyle p}in the prime factorization ofn!{\displaystyle n!}as∑i=1∞⌊npi⌋=n−sp(n)p−1.{\displaystyle \sum _{i=1}^{\infty }\left\lfloor {\frac {n}{p^{i}}}\right\rfloor ={\frac {n-s_{p}(n)}{p-1}}.}Heresp(n){\displaystyle s_{p}(n)}denotes the sum of thebase-p{\displaystyle p}digitsofn{\displaystyle n},and the exponent given by this formula can also be interpreted in advanced mathematics as thep-adic valuationof the factorial.Applying Legendre's formula to the product formula forbinomial coefficientsproducesKummer's theorem, a similar result on the exponent of each prime in the factorization of a binomial coefficient.Grouping the prime factors of the factorial intoprime powersin different ways produces themultiplicative partitions of factorials.
The special case of Legendre's formula forp=5{\displaystyle p=5}gives the number oftrailing zerosin the decimal representation of the factorials.According to this formula, the number of zeros can be obtained by subtracting the base-5 digits ofn{\displaystyle n}fromn{\displaystyle n}, and dividing the result by four.Legendre's formula implies that the exponent of the primep=2{\displaystyle p=2}is always larger than the exponent forp=5{\displaystyle p=5},so each factor of five can be paired with a factor of two to produce one of these trailing zeros.The leading digits of the factorials are distributed according toBenford's law.Every sequence of digits, in any base, is the sequence of initial digits of some factorial number in that base.
Another result on divisibility of factorials,Wilson's theorem, states that(n−1)!+1{\displaystyle (n-1)!+1}is divisible byn{\displaystyle n}if and only ifn{\displaystyle n}is aprime number.For any givenintegerx{\displaystyle x},theKempner functionofx{\displaystyle x}is given by the smallestn{\displaystyle n}for whichx{\displaystyle x}dividesn!{\displaystyle n!}.For almost all numbers (all but a subset of exceptions withasymptotic densityzero), it coincides with the largest prime factorofx{\displaystyle x}.
The product of two factorials,m!⋅n!{\displaystyle m!\cdot n!},always evenly divides(m+n)!{\displaystyle (m+n)!}.There are infinitely many factorials that equal the product of other factorials: ifn{\displaystyle n}is itself any product of factorials, thenn!{\displaystyle n!}equals that same product multiplied by one more factorial,(n−1)!{\displaystyle (n-1)!}.The only known examples of factorials that are products of other factorials but are not of this "trivial" form are9!=7!⋅3!⋅3!⋅2!{\displaystyle 9!=7!\cdot 3!\cdot 3!\cdot 2!},10!=7!⋅6!=7!⋅5!⋅3!{\displaystyle 10!=7!\cdot 6!=7!\cdot 5!\cdot 3!},and16!=14!⋅5!⋅2!{\displaystyle 16!=14!\cdot 5!\cdot 2!}.It would follow from theabcconjecturethat there are only finitely many nontrivial examples.
Thegreatest common divisorof the values of aprimitive polynomialof degreed{\displaystyle d}over the integers evenly dividesd!{\displaystyle d!}.
There are infinitely many ways to extend the factorials to acontinuous function.The most widely used of theseuses thegamma function, which can be defined for positive real numbers as theintegralΓ(z)=∫0∞xz−1e−xdx.{\displaystyle \Gamma (z)=\int _{0}^{\infty }x^{z-1}e^{-x}\,dx.}The resulting function is related to the factorial of a non-negative integern{\displaystyle n}by the equationn!=Γ(n+1),{\displaystyle n!=\Gamma (n+1),}which can be used as a definition of the factorial for non-integer arguments.
At all valuesx{\displaystyle x}for which bothΓ(x){\displaystyle \Gamma (x)}andΓ(x−1){\displaystyle \Gamma (x-1)}are defined, the gamma function obeys thefunctional equationΓ(n)=(n−1)Γ(n−1),{\displaystyle \Gamma (n)=(n-1)\Gamma (n-1),}generalizing therecurrence relationfor the factorials.
The same integral converges more generally for anycomplex numberz{\displaystyle z}whose real part is positive. It can be extended to the non-integer points in the rest of thecomplex planeby solving for Euler'sreflection formulaΓ(z)Γ(1−z)=πsin⁡πz.{\displaystyle \Gamma (z)\Gamma (1-z)={\frac {\pi }{\sin \pi z}}.}However, this formula cannot be used at integers because, for them, thesin⁡πz{\displaystyle \sin \pi z}term would produce adivision by zero. The result of this extension process is ananalytic function, theanalytic continuationof the integral formula for the gamma function. It has a nonzero value at all complex numbers, except for the non-positive integers where it hassimple poles. Correspondingly, this provides a definition for the factorial at all complex numbers other than the negative integers.One property of the gamma function, distinguishing it from other continuous interpolations of the factorials, is given by theBohr–Mollerup theorem, which states that the gamma function (offset by one) is the onlylog-convexfunction on the positive real numbers that interpolates the factorials and obeys the same functional equation. A related uniqueness theorem ofHelmut Wielandtstates that the complex gamma function and its scalar multiples are the onlyholomorphic functionson the positive complex half-plane that obey the functional equation and remain bounded for complex numbers with real part between 1 and 2.
Other complex functions that interpolate the factorial values includeHadamard's gamma function, which is anentire functionover all the complex numbers, including the non-positive integers.In thep-adic numbers, it is not possible to continuously interpolate the factorial function directly, because the factorials of large integers (a dense subset of thep-adics) converge to zero according to Legendre's formula, forcing any continuous function that is close to their values to be zero everywhere. Instead, thep-adic gamma functionprovides a continuous interpolation of a modified form of the factorial, omitting the factors in the factorial that are divisible byp.
Thedigamma functionis thelogarithmic derivativeof the gamma function. Just as the gamma function provides a continuous interpolation of the factorials, offset by one, the digamma function provides a continuous interpolation of theharmonic numbers, offset by theEuler–Mascheroni constant.
क्रमगुणित फलनवैज्ञानिक कैलकुलेटरमें एक सामान्य विशेषता है।इसे वैज्ञानिक प्रोग्रामिंग लाइब्रेरीज़ जैसे किपायथनगणितीय फलन मॉड्यूल में भी शामिल किया गया है।and theBoost C++ library.If efficiency is not a concern, computing factorials is trivial: just successively multiply a variable initializedto1{\displaystyle 1}by the integers upton{\displaystyle n}.The simplicity of this computation makes it a common example in the use of different computer programming styles and methods.
The computation ofn!{\displaystyle n!}can be expressed inpseudocodeusingiterationas
or usingrecursionbased on its recurrence relation as
Other methods suitable for its computation includememoization,dynamic programming,andfunctional programming.Thecomputational complexityof these algorithms may be analyzed using the unit-costrandom-access machinemodel of computation, in which each arithmetic operation takes constant time and each number uses a constant amount of storage space. In this model, these methods can computen!{\displaystyle n!}in timeO(n){\displaystyle O(n)},and the iterative version uses spaceO(1){\displaystyle O(1)}.Unless optimized fortail recursion, the recursive version takes linear space to store itscall stack.However, this model of computation is only suitable whenn{\displaystyle n}is small enough to allown!{\displaystyle n!}to fit into amachine word.The values 12! and 20! are the largest factorials that can be stored in, respectively, the32-bitand64-bitintegers.Floating pointcan represent larger factorials, but approximately rather than exactly, and will still overflow for factorials larger than170!{\displaystyle 170!}.
The exact computation of larger factorials involvesarbitrary-precision arithmetic, because offast growthandinteger overflow.  Time of computation can be analyzed as a function of the number of digits or bits in the result.By Stirling's formula,n!{\displaystyle n!}hasb=O(nlog⁡n){\displaystyle b=O(n\log n)}bits.TheSchönhage–Strassen algorithmcan produce ab{\displaystyle b}-bitproduct in timeO(blog⁡blog⁡log⁡b){\displaystyle O(b\log b\log \log b)},and fastermultiplication algorithmstaking timeO(blog⁡b){\displaystyle O(b\log b)}are known.However, computing the factorial involves repeated products, rather than a single multiplication, so these time bounds do not apply directly.  In this setting, computingn!{\displaystyle n!}by multiplying the numbers from 1ton{\displaystyle n}in sequence is inefficient, because it involvesn{\displaystyle n}multiplications, a constant fraction of which take timeO(nlog2⁡n){\displaystyle O(n\log ^{2}n)}each, giving total timeO(n2log2⁡n){\displaystyle O(n^{2}\log ^{2}n)}.A better approach is to perform the multiplications as adivide-and-conquer algorithmthat multiplies a sequence ofi{\displaystyle i}numbers by splitting it into two subsequences ofi/2{\displaystyle i/2}numbers, multiplies each subsequence, and combines the results with one last multiplication. This approach to the factorial takes total timeO(nlog3⁡n){\displaystyle O(n\log ^{3}n)}:one logarithm comes from the number of bits in the factorial, a second comes from the multiplication algorithm, and a third comes from the divide and conquer.
Even better efficiency is obtained by computingn!from its prime factorization, based on the principle thatexponentiation by squaringis faster than expanding an exponent into a product.An algorithm for this byArnold Schönhagebegins by finding the list of the primes upton{\displaystyle n},for instance using thesieve of Eratosthenes, and uses Legendre's formula to compute the exponent for each prime. Then it computes the product of the prime powers with these exponents, using a recursive algorithm, as follows:
The product of all primes up ton{\displaystyle n}is anO(n){\displaystyle O(n)}-bit number, by theprime number theorem, so the time for the first step isO(nlog2⁡n){\displaystyle O(n\log ^{2}n)}, with one logarithm coming from the divide and conquer and another coming from the multiplication algorithm. In the recursive calls to the algorithm, the prime number theorem can again be invoked to prove that the numbers of bits in the corresponding products decrease by a constant factor at each level of recursion, so the total time for these steps at all levels of recursion adds in ageometric seriestoO(nlog2⁡n){\displaystyle O(n\log ^{2}n)}.The time for the squaring in the second step and the multiplication in the third step are againO(nlog2⁡n){\displaystyle O(n\log ^{2}n)},because each is a single multiplication of a number withO(nlog⁡n){\displaystyle O(n\log n)}bits. Again, at each level of recursion the numbers involved have a constant fraction as many bits (because otherwise repeatedly squaring them would produce too large a final result) so again the amounts of time for these steps in the recursive calls add in a geometric seriestoO(nlog2⁡n){\displaystyle O(n\log ^{2}n)}.Consequentially, the whole algorithm takestimeO(nlog2⁡n){\displaystyle O(n\log ^{2}n)},proportional to a single multiplication with the same number of bits in its result.
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