11 Big Integers
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\( \newcommand {\multicolumn }[3]{#3}\)
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\(\newcommand {\idiv }[2]{\lfloor #1/#2\rfloor }\)
\(\newcommand \attribdot {\ensuremath {\mkern 1.5mu.\mkern 1.5mu}}\)
\(\newcommand \attribxr [2]{#1\attribdot \text {#2}}\)
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\(\newcommand \attribii [2]{\Id {#1}\attribdot \Id {#2}}\)
\(\newcommand \textsc [1]{#1}\)
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\(\renewcommand {\rowcolor }[2][named]{\LWRorigrowcolor [#1]{#2}\LWRabsorbtwooptions }\)
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\(\newcommand {\tcbset }[1]{}\)
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\(\let \symbbit \mathbb \)
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\(\newcommand {\mathfence }[1]{\mathord {#1}}\)
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\(\def\Alpha{\unicode{x1D6E2}}\)
\(\def\Beta{\unicode{x1D6E3}}\)
\(\def\Gamma{\unicode{x1D6E4}}\)
\(\def\Digamma{\mathit{\unicode{x03DC}}}\)
\(\def\Delta{\unicode{x1D6E5}}\)
\(\def\Epsilon{\unicode{x1D6E6}}\)
\(\def\Zeta{\unicode{x1D6E7}}\)
\(\def\Eta{\unicode{x1D6E8}}\)
\(\def\Theta{\unicode{x1D6E9}}\)
\(\def\Vartheta{\unicode{x1D6F3}}\)
\(\def\Iota{\unicode{x1D6EA}}\)
\(\def\Kappa{\unicode{x1D6EB}}\)
\(\def\Lambda{\unicode{x1D6EC}}\)
\(\def\Mu{\unicode{x1D6ED}}\)
\(\def\Nu{\unicode{x1D6EE}}\)
\(\def\Xi{\unicode{x1D6EF}}\)
\(\def\Omicron{\unicode{x1D6F0}}\)
\(\def\Pi{\unicode{x1D6F1}}\)
\(\def\Rho{\unicode{x1D6F2}}\)
\(\def\Sigma{\unicode{x1D6F4}}\)
\(\def\Tau{\unicode{x1D6F5}}\)
\(\def\Upsilon{\unicode{x1D6F6}}\)
\(\def\Phi{\unicode{x1D6F7}}\)
\(\def\Chi{\unicode{x1D6F8}}\)
\(\def\Psi{\unicode{x1D6F9}}\)
\(\def\Omega{\unicode{x1D6FA}}\)
\(\def\alpha{\unicode{x1D6FC}}\)
\(\def\beta{\unicode{x1D6FD}}\)
\(\def\varbeta{\unicode{x03D0}}\)
\(\def\gamma{\unicode{x1D6FE}}\)
\(\def\digamma{\mathit{\unicode{x03DD}}}\)
\(\def\delta{\unicode{x1D6FF}}\)
\(\def\epsilon{\unicode{x1D716}}\)
\(\def\varepsilon{\unicode{x1D700}}\)
\(\def\zeta{\unicode{x1D701}}\)
\(\def\eta{\unicode{x1D702}}\)
\(\def\theta{\unicode{x1D703}}\)
\(\def\vartheta{\unicode{x1D717}}\)
\(\def\iota{\unicode{x1D704}}\)
\(\def\kappa{\unicode{x1D705}}\)
\(\def\varkappa{\unicode{x1D718}}\)
\(\def\lambda{\unicode{x1D706}}\)
\(\def\mu{\unicode{x1D707}}\)
\(\def\nu{\unicode{x1D708}}\)
\(\def\xi{\unicode{x1D709}}\)
\(\def\omicron{\unicode{x1D70A}}\)
\(\def\pi{\unicode{x1D70B}}\)
\(\def\varpi{\unicode{x1D71B}}\)
\(\def\rho{\unicode{x1D70C}}\)
\(\def\varrho{\unicode{x1D71A}}\)
\(\def\sigma{\unicode{x1D70E}}\)
\(\def\varsigma{\unicode{x1D70D}}\)
\(\def\tau{\unicode{x1D70F}}\)
\(\def\upsilon{\unicode{x1D710}}\)
\(\def\phi{\unicode{x1D719}}\)
\(\def\varphi{\unicode{x1D711}}\)
\(\def\chi{\unicode{x1D712}}\)
\(\def\psi{\unicode{x1D713}}\)
\(\def\omega{\unicode{x1D714}}\)
\(\def\upAlpha{\unicode{x0391}}\)
\(\def\upBeta{\unicode{x0392}}\)
\(\def\upGamma{\unicode{x0393}}\)
\(\def\upDigamma{\unicode{x03DC}}\)
\(\def\upDelta{\unicode{x0394}}\)
\(\def\upEpsilon{\unicode{x0395}}\)
\(\def\upZeta{\unicode{x0396}}\)
\(\def\upEta{\unicode{x0397}}\)
\(\def\upTheta{\unicode{x0398}}\)
\(\def\upVartheta{\unicode{x03F4}}\)
\(\def\upIota{\unicode{x0399}}\)
\(\def\upKappa{\unicode{x039A}}\)
\(\def\upLambda{\unicode{x039B}}\)
\(\def\upMu{\unicode{x039C}}\)
\(\def\upNu{\unicode{x039D}}\)
\(\def\upXi{\unicode{x039E}}\)
\(\def\upOmicron{\unicode{x039F}}\)
\(\def\upPi{\unicode{x03A0}}\)
\(\def\upVarpi{\unicode{x03D6}}\)
\(\def\upRho{\unicode{x03A1}}\)
\(\def\upSigma{\unicode{x03A3}}\)
\(\def\upTau{\unicode{x03A4}}\)
\(\def\upUpsilon{\unicode{x03A5}}\)
\(\def\upPhi{\unicode{x03A6}}\)
\(\def\upChi{\unicode{x03A7}}\)
\(\def\upPsi{\unicode{x03A8}}\)
\(\def\upOmega{\unicode{x03A9}}\)
\(\def\itAlpha{\unicode{x1D6E2}}\)
\(\def\itBeta{\unicode{x1D6E3}}\)
\(\def\itGamma{\unicode{x1D6E4}}\)
\(\def\itDigamma{\mathit{\unicode{x03DC}}}\)
\(\def\itDelta{\unicode{x1D6E5}}\)
\(\def\itEpsilon{\unicode{x1D6E6}}\)
\(\def\itZeta{\unicode{x1D6E7}}\)
\(\def\itEta{\unicode{x1D6E8}}\)
\(\def\itTheta{\unicode{x1D6E9}}\)
\(\def\itVartheta{\unicode{x1D6F3}}\)
\(\def\itIota{\unicode{x1D6EA}}\)
\(\def\itKappa{\unicode{x1D6EB}}\)
\(\def\itLambda{\unicode{x1D6EC}}\)
\(\def\itMu{\unicode{x1D6ED}}\)
\(\def\itNu{\unicode{x1D6EE}}\)
\(\def\itXi{\unicode{x1D6EF}}\)
\(\def\itOmicron{\unicode{x1D6F0}}\)
\(\def\itPi{\unicode{x1D6F1}}\)
\(\def\itRho{\unicode{x1D6F2}}\)
\(\def\itSigma{\unicode{x1D6F4}}\)
\(\def\itTau{\unicode{x1D6F5}}\)
\(\def\itUpsilon{\unicode{x1D6F6}}\)
\(\def\itPhi{\unicode{x1D6F7}}\)
\(\def\itChi{\unicode{x1D6F8}}\)
\(\def\itPsi{\unicode{x1D6F9}}\)
\(\def\itOmega{\unicode{x1D6FA}}\)
\(\def\upalpha{\unicode{x03B1}}\)
\(\def\upbeta{\unicode{x03B2}}\)
\(\def\upvarbeta{\unicode{x03D0}}\)
\(\def\upgamma{\unicode{x03B3}}\)
\(\def\updigamma{\unicode{x03DD}}\)
\(\def\updelta{\unicode{x03B4}}\)
\(\def\upepsilon{\unicode{x03F5}}\)
\(\def\upvarepsilon{\unicode{x03B5}}\)
\(\def\upzeta{\unicode{x03B6}}\)
\(\def\upeta{\unicode{x03B7}}\)
\(\def\uptheta{\unicode{x03B8}}\)
\(\def\upvartheta{\unicode{x03D1}}\)
\(\def\upiota{\unicode{x03B9}}\)
\(\def\upkappa{\unicode{x03BA}}\)
\(\def\upvarkappa{\unicode{x03F0}}\)
\(\def\uplambda{\unicode{x03BB}}\)
\(\def\upmu{\unicode{x03BC}}\)
\(\def\upnu{\unicode{x03BD}}\)
\(\def\upxi{\unicode{x03BE}}\)
\(\def\upomicron{\unicode{x03BF}}\)
\(\def\uppi{\unicode{x03C0}}\)
\(\def\upvarpi{\unicode{x03D6}}\)
\(\def\uprho{\unicode{x03C1}}\)
\(\def\upvarrho{\unicode{x03F1}}\)
\(\def\upsigma{\unicode{x03C3}}\)
\(\def\upvarsigma{\unicode{x03C2}}\)
\(\def\uptau{\unicode{x03C4}}\)
\(\def\upupsilon{\unicode{x03C5}}\)
\(\def\upphi{\unicode{x03D5}}\)
\(\def\upvarphi{\unicode{x03C6}}\)
\(\def\upchi{\unicode{x03C7}}\)
\(\def\uppsi{\unicode{x03C8}}\)
\(\def\upomega{\unicode{x03C9}}\)
\(\def\italpha{\unicode{x1D6FC}}\)
\(\def\itbeta{\unicode{x1D6FD}}\)
\(\def\itvarbeta{\unicode{x03D0}}\)
\(\def\itgamma{\unicode{x1D6FE}}\)
\(\def\itdigamma{\mathit{\unicode{x03DD}}}\)
\(\def\itdelta{\unicode{x1D6FF}}\)
\(\def\itepsilon{\unicode{x1D716}}\)
\(\def\itvarepsilon{\unicode{x1D700}}\)
\(\def\itzeta{\unicode{x1D701}}\)
\(\def\iteta{\unicode{x1D702}}\)
\(\def\ittheta{\unicode{x1D703}}\)
\(\def\itvartheta{\unicode{x1D717}}\)
\(\def\itiota{\unicode{x1D704}}\)
\(\def\itkappa{\unicode{x1D705}}\)
\(\def\itvarkappa{\unicode{x1D718}}\)
\(\def\itlambda{\unicode{x1D706}}\)
\(\def\itmu{\unicode{x1D707}}\)
\(\def\itnu{\unicode{x1D708}}\)
\(\def\itxi{\unicode{x1D709}}\)
\(\def\itomicron{\unicode{x1D70A}}\)
\(\def\itpi{\unicode{x1D70B}}\)
\(\def\itvarpi{\unicode{x1D71B}}\)
\(\def\itrho{\unicode{x1D70C}}\)
\(\def\itvarrho{\unicode{x1D71A}}\)
\(\def\itsigma{\unicode{x1D70E}}\)
\(\def\itvarsigma{\unicode{x1D70D}}\)
\(\def\ittau{\unicode{x1D70F}}\)
\(\def\itupsilon{\unicode{x1D710}}\)
\(\def\itphi{\unicode{x1D719}}\)
\(\def\itvarphi{\unicode{x1D711}}\)
\(\def\itchi{\unicode{x1D712}}\)
\(\def\itpsi{\unicode{x1D713}}\)
\(\def\itomega{\unicode{x1D714}}\)
\(\let \lparen (\)
\(\let \rparen )\)
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\(\newcommand {\intcap }{\mathop {\unicode {x2A19}}\limits }\)
\(\newcommand {\intcup }{\mathop {\unicode {x2A1A}}\limits }\)
\(\newcommand {\upint }{\mathop {\unicode {x2A1B}}\limits }\)
\(\newcommand {\lowint }{\mathop {\unicode {x2A1C}}\limits }\)
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\(\newcommand {\bigtalloblong }{\mathop {\unicode {x2AFF}}\limits }\)
\(\newcommand {\arabicmaj }{\mathop {\unicode {x1EEF0}}\limits }\)
\(\newcommand {\arabichad }{\mathop {\unicode {x1EEF1}}\limits }\)
11.10 Chapter Notes¶
The topics we discussed in this chapter belong to the field of computer arithmetic, which studies algorithms and data structures for computing with different kinds of numbers. Computer scientists have come up with a surprising number of data types for representing numbers:
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• Fixed-width integers are the familiar “computer integers” with a fixed number of bits. We discussed some of their properties in Chapter 5.
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• Big integers, also known as bignums, can grow to any size, provided there is sufficient memory. We discussed the basic algorithms for working with big integers in this chapter.
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• Integers modulo \(m\) represent numbers in the range \([0\isep m-1]\); arithmetic operations are defined to wrap around at either end of the range. Such numbers are widely used in number theoretic applications, including cryptography.
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• Fractions are quotients of form \(n/m\), where \(n\) and \(m\) are two integers. The numerical range of fractions depends on the integer type that is used to store the numerator \(n\) and the denominator \(m\).
Arbitrary-precision fractions, where \(n\) and \(m\) are big integers, correspond to rational numbers in mathematics and are integral to the functioning of computer algebra systems.
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• Fixed-point numbers are fractions of the form \(n/2^K\), where the numerator \(n\) is a fixed-width integer and the denominator \(2^K\) is a fixed power of 2. For instance, when \(n\) is a
32-bit integer and \(K=16\), this representation lets us store numbers between \(-2^{31}/2^{16}=-32768\) and \((2^{31}-1)/2^{16}\approx 32767.99998\). The main advantages of fixed-point numbers are that only denominator \(n\) needs to be stored and that arithmetic operations are easy to
implement.
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• Floating-point numbers are numbers of the form \(n\cdot 2^e\), where both \(n\) and \(e\) are fixed-width integers. Since the exponent \(e\) is variable, floating point numbers can represent
both large and tiny numbers, while still taking up only a fixed number of bits. Floating-point numbers strike a good balance between the flexibility of fractions and the efficiency of fixed-point numbers, but they are much harder to implement than either.
Two good books on computer arithmetic are Modern Computer Arithmetic by Brent and Zimmerman [20] and Volume 2 of The Art of Computer Programming [60]. For floating-point numbers in particular, see the books by Overton [74] and Muller et al. [71].
If you are interested in the implementation of big integers, you may be interested in studying one of the many publicly available implementations. The source code of the BigInteger in Java’s standard library (http://hg.openjdk.java.net/jdk8/jdk8/jdk/file/tip/src/share/classes/java/math/BigInteger.java) is fairly easy to read and implements several advanced
arithmetic algorithms. One of the fastest libraries for arbitrary-precision arithmetic is the GNU Multiple Precision Arithmetic Library (GMP) [44].
Over the course of history, mathematicians have come up with many different multiplication algorithms to simplify the nontrivial task of computing the product of two numbers [37]. The ancient Egyptians used a method based on repeated doubling, and a similar method, known
as Russian peasant multiplication, is based on repeated halving of one argument and doubling of the other. Our modern table-based algorithm originated in India and evolved into its current form as the Hindu-Arabic numerals (the decimal numbers
we now take for granted) slowly took hold in Europe between 1200 and 1800 CE.
For many centuries this method remained the standard algorithm for multiplying numbers. Around 1956, the Soviet mathematician Andrey Kolmogorov conjectured that it is essentially optimal in the sense that no algorithm for multiplying two \(n\)-digit numbers could improve on its runtime complexity
of \(O(n^2)\). It took only a few years for this conjecture to be disproved, when Anatoly Karatsuba, one of his Kolmogorov’s own students, came up with the \(O(n^{1.58})\) algorithm discussed in Section 11.5 [57]. This was clearly a major breakthrough, but it also raised two tantalizing questions: Are there even faster multiplication algorithms? Or maybe even
a fastest algorithm?
The first improvement on Karatsuba’s method was found independently by Andrei Toom in 1963 [92] and by Stephen Cook in 1966 [24, Chapter 3]. The resulting Toom-Cook algorithm is a relatively simple variation of Karatsuba’s method that improves its runtime complexity to \(O(n^{1.46})\). In 1971, Arnold Schönhage and Volker Strassen discovered an even better algorithm that is based on the fast
Fourier transform. Schönhage-Strassen multiplication requires only \(O(n\log n\cdot \log (\log n))\) steps. Since the \(\log (\log n)\) term grows very slowly, Schönhage-Strassen multiplication behaves almost like a
\(O(n\log n)\) algorithm and is currently the best practical algorithm for multiplying large integers with thousands of digits.
After Schönhage and Strassen demonstrated that multiplication is almost \(O(n\log n)\), mathematicians started to search for an multiplication method that truly has this complexity. This search took many years and several intermediate steps. In 2007, Martin Fürer first
demonstrated how to reduce the \(\log (\log n)\) factor of Schönhage-Strassen multiplication by tweaking the implementation of the fast Fourier transform. Extending this result, in 2019 David Harvey and Joris van der Hoeven finally
managed to get rid of the factor altogether to obtain the first multiplication algorithm that requires only \(O(n\log n)\) operations [48, 59].
It is currently conjectured that no multiplication algorithm exists with a better runtime complexity than \(O(n\log n)\). It should be noted, however, that for now the Harvey-van der Hoeven algorithm is mainly of theoretical interest. By their own estimate, the algorithm is
faster than Schönhage and Strassen’s only for truly astronomical numbers with more than \(2^{1729^{12}}\) bits!
The number \(\pi \) has a long and storied history that stretches across several millennia and multiple countries. The earliest known approximations to \(\pi \) date back to the ancient Babylonians and Egyptians. A clay tablet found near the city of Susa (dated
between 1900 and 1700 BCE) indicates that Babylonian mathematicians approximated \(\pi \) as \(25/8\approx 3.125\), and the Rhind Papyrus, an Egyptian collection of mathematical problems and their solutions (dated to around 1650 BCE) uses the approximation \(256/81\approx 3.16\).
The first real algorithms for computing \(\pi \) were developed by the Greek mathematician Archimedes in the 3rd century BCE and independently by the Chinese mathematician Liu Hui in 263 CE. As discussed in Exercise 11.22, Archimedes’ method is based on the idea of fitting polygons to a circle and makes it possible to compute \(\pi \) to a few decimal digits. The method converges slowly, however, and the required computations become increasingly arduous.
It wasn’t until the beginning of the 15th century that the Persian mathematician Jamshīd al-Kāshī finally managed to compute 16 decimal digits of \(\pi \), and in 1600 Ludolph van Ceulen attained a remarkable 35 decimal digits, which corresponds to approximating the circle by \(2.8\times
10^{19}\)-sided polygons! Performing the necessary computations took up a major part of van Ceulen’s life.
The next era in \(\pi \)’s history begins with discovery of calculus in the 17th century and the subsequent development of powerful computational methods such as power series, derivatives, and integrals. Arguably the most important formula for computing \(\pi \) from this era is Machin’s formula, which was discovered by the English astronomer John Machin in 1706:
\(\seteqnumber{0}{11.}{25}\)
\begin{equation*}
\frac {\pi }{4}=4\arctan \frac {1}{5}-\arctan \frac {1}{239}.
\end{equation*}
Machin himself used this formula to compute a record-breaking 100 digits of \(\pi \), and it remained the most efficient method far into the 20th century. In fact, Machin’s formula was used in the very first computer-based computation of \(\pi \): In 1948, George Reitwiesner, John von
Neumann, and N.C. Metropolis used it to compute 2037 digits of \(\pi \) on the ENIAC, one of the earliest electronic computers.
With the progress of computer technology after WWII, new \(\pi \) records were now set every few years. As of 2025, 300 trillion decimal digits of \(\pi \) have been computed. All of the recent world records in computing \(\pi \) were achieved using a formula for \(\pi \) that was discovered in
1989 by Gregory and David Chudnovsky:
\(\seteqnumber{0}{11.}{25}\)
\begin{equation*}
\frac {1}{\pi } = 12\sum _{k=0}^\infty \frac {(-1)^k(6k)!(13591409+545140134k)} {(3k)!(k!)^3640320^{3k+3/2}}
\end{equation*}
This Chudnovsky formula converges even more rapidly than Machin’s formula and, despite its daunting appearance, can be implemented very efficiently.
For a more detailed history of \(\pi \) and its computation, see the paper by Bailey et al. [9]. A detailed account of the various formulas and algorithms for computing \(\pi \) that have been devised over the centuries can be found in
the book by Arndt and Haenel [6]. As of June 2025, the most advanced program for computing \(\pi \) and many other mathematical constants is y-cruncher [98].