Primary: knuth1963 [Knu63], leveque1956 [LeV56, chapter 3], ribenboim1979 [Rib79], wiles1995 [Wil95]; Secondary: bell1990 [Bel90], euclid2006 [Euc06], gauss1965 [Gau65], heath2008 [Hea08], singh1998 [Sin98].
Primary: floyd1994 [FB94, chapter 3.3], markov1954 [Mar54], markov1988 [MN88], zemanek1981 [Zem81]. Secondary: church1985 [Chu85], church1996 [Chu96], church2019 [Chu19], copeland2004 [Cop04], copeland2013 [CPS13], davis2004 [Dav04], godel2001a [Göd01a], godel2001b [Göd01b], godel2001c [Göd01c], godel2013a [Göd13a], godel2013b [Göd13b], hoare1972 [HA72], hofstadter1999 [Hof99], kleene2012 [Kle12], vanheijenoort2002 [vH02], whitehead2011 [WR11].
Primary: graham1994 [GKP94].
Primary: bussey1917 [Bus17], cajori1918 [Caj18], conway1996 [CG96, chapter 2], dijkstra1975 [Dij75], dijkstra1976 [Dij76], floyd1967 [Flo67], floyd1972 [Flo72], gardner1986 [Gar86, chapter 16], golomb1965 [Gol65], hardy1999 [Har99, chapters 6, 8], henkin1960 [Hen60], hoare1971 [Hoa71], king1972 [Kin72], morris1984 [MJ84], naur1966 [Nau66], polya1954 [Pól54, chapter 7], rashed1972 [Ras72], sduk1838 [Soc38, pp. 465–466], turing1949 [Tur49], vonneumann1963 [vN63, pp. 91–99]. Secondary: conway2001 [Con01], fraleigh2002 [Fra02], hardy2008 [HW08].
Primary: cajori1913a [Caj13a], cajori1913b [Caj13b], cajori1913c [Caj13c], cajori1913d [Caj13d], cajori1913e [Caj13e], cajori1913f [Caj13f], cajori1913g [Caj13g], chen1972 [Che72], linskii1957 [Lin57], knuth1986d [Knu86c, § 120–147], meggitt1962 [Meg62], meggitt1963 [Meg63]. Secondary: feynman1999 [Fey99], hamming1997 [Ham97], hey2002 [Hey02], knuth1986a [Knu86e], knuth1986b [Knu86d], knuth1986c [Knu86b], knuth1986e [Knu86a].
Primary: apostol1974 [Apo74, 1st edition: chapter 12; or 2nd edition: chapter 8], binet1813 [Bin13], cauchy1815 [Cau15], cauchy1841 [Cau41], chebyshev1882 [Che82], demoivre1738 [dM38, pp. 197–199], euler1827a [Eul27a, § 1169], knuth1992 [Knu92], krattenthaler1990 [Kra90], sylvester1857 [Syl57], todd1961 [Tod61], waring1779 [War79]. Secondary: cauchy1840 [Cau40, pp. 151–159], cauchy1844 [Cau44], cauchy1847 [Cau47], euler1824 [Eul24], euler1827b [Eul27b], euler1845 [Eul45], krattenthaler1988 [Kra88], livio2003 [Liv03].
Primary: busche1909 [Bus09], eisele1990 [EH90], eisenstein1844 [Eis44], hamel1905 [Ham05], hermite1884 [Her84], mordell1958 [Mor58], yoder1975 [Yod75].
Primary: arbogast1800 [Arb00, § 52], capelli1893 [Cap93], davis1959 [Dav59], dutka1984 [Dut84], faadibruno1857 [dB57], good1961 [Goo61], hankel1864 [Han64], knopp1951 [Kno51, pp. 518–520, 527, 534], kramp1808 [Kra08, p. 219], legendre1808 [Leg08, p. 8], stirling1753 [Sti53, p. 137], struik1969 [Str69, pp. 244–253], toscano1939 [Tos39], uhler1955 [Uhl55], wall1938 [Wal38]. Secondary: euler1729 [Eul29], hermite1917 [Her17, proof of 1900, pp. 529–543], kramp1801 [Kra01], shanks1962 [SW62], smith1958a [Smi58a], smith1958b [Smi58b], stickelberger1890 [Sti90], tweddle2003 [Twe03], vacca1899 [Vac99], waring1782 [War82, 1st edition: p. 218; or 3rd edition: p. 380], weeks1991 [Wee91].
Primary: abel1826a [Abe26a], barton1960 [BDM60], barton1963 [BDM63], cauchy1843 [Cau43], chakravarti1932 [Cha32], demoivre1730 [dM30, p. 101], diaconis1991 [DZ91], dickson1902 [Dic02], dixon1891 [Dix91], dixon1903 [Dix03], dougall1907 [Dou07], dutka1984 [Dut84], euler1774 [Eul74], fine1947 [Fin47], fostemann1835 [Fös35], gasper1990 [GR90], gauss1813 [Gau13], gauss1808 [Gau08], gould1956 [Gou56], gould1962 [SMG62, p. 572], gould1966 [GK66], graham1994 [GKP94, § 5.5, 5.6, 5.8, 6.1, 6.2, 6.5; exercises 5.75, 5.83, 5.106, 6.57], heine1847 [Hei47], hui1407 [Hui07, chapter 16,344], hurwitz1902 [Hur02], hwang1995 [Hwa95], jacobi1846 [Jac46], karamata1935 [Kar35], knuth1971 [Knu71], knuth1989 [KW89], knuth1992 [Knu92, pp. 403–422; pp. 410–422], kramp1808 [Kra08, p. 359], kummer1852 [Kum52], lovasz1993 [Lov93, problem 13.31], lucas1878b [Luc78b, pp. 229–230], macmahon1902 [Mac02], mahavira1912 [Mah12, chapter 6], moser1958a [MW58b], moser1958b [MW58a], needham1959 [Nee59, pp. 138-139], newton1676a [New76a], newton1676b [New76b], pascal1665 [Pas65], pascal1887 [Pas87], patwardhan2006 [PNS06, § CXX, CXXII], petkovsek1996 [PWZ96], pfaff1797 [Pfa97, pp. 38–57], pingala1874 [Piṅ74], poincare1896 [Poi96, pp. 56-60], ramshaw1977 [Ram77], ramus1834 [Ram34], redheffer1996 [Red96], rogers1895 [Rog95, pp. 15–32; § 8], rothe1793 [Rot93], rothe1811 [Rot11, p. xxix], scherk1828 [Sch28], schweins1820 [Sch20, § 151], stirling1753 [Sti53], temme1993 [Tem93], torelli1895 [Tor95], vandermonde1772 [Van72], vonettingshausen1826 [vE26, § 31], wilf1993 [Wil93]. Secondary: alder2002 [Ald02], logan1987 [Log87], lucas1878a [Luc78a], lucas1878c [Luc78c], needham1954 [Nee54], needham1956 [Nee56].
Primary: cullen2007 [Cul07], euler1734 [Eul34], gould1961 [Gou61], graham1994 [GKP94, § 6.5], hardy2008 [HW08, section 7.8], klamkin1962 [KKWvL62], nanjundiah1963 [Nan63], selfridge1960 [SS60]. Secondary: waring1782 [War82, 2nd edition: p. 383].
Primary: bergman1957 [Ber57], bernoulli1728 [Ber28, § 7], boncampagni1857 [Bon57, pp. 283–285], conway1996 [CG96, pp. 113–126], cook1979 [Coo79, p. 420], coxeter1953 [Cox53], demoivre1722 [dM22], dickson2005a [Dic05a, chapter 17], gardner1961 [Gar61, chapter 8], graham1994 [GKP94, § 6.6], jarden1966 [Jar66, pp. 30–33], kepler1608 [Kep08], kepler2014 [Kep14, p. 21], klarner1968 [Kla68], lekkerkerker1952 [Lek52], lucas1878b [Luc78b, pp. 201–204], markowsky1992 [Mar92], pisano1202 [Pis02], riordan1962 [Rio62], schreiber1995 [Sch95], singh1985 [Sin85], stolarsky1976 [Sto76], schwenk1970 [Sch70], whinihan1963 [Whi63]. Secondary: bernoulliiii1771 [III72], boncampagni1862 [Bon62], cooper1995 [CK95], dickson2005b [Dic05b], dickson2005c [Dic05c], frey1947 [Fre47, pp. 705–706], gardner2008a [Gar08b, chapter 8], gardner2008b [Gar08a], gardner2009 [Gar09], gardner2014 [Gar14], jacob1565 [Jac65], jarden1949 [JM49], kepler1955 [Kep55, pp. 154–165,429] leger1837 [Lég37], lucas1876a [Luc76a], lucas1876b [Luc76b], markov1882 [Mar82], matiyasevich1993 [Mat93], shallit1994 [Sha94], sigler2003 [Pis03, pp. 404–405], vorobev2002 [Vor02], zeckendorf1972 [Zec72].
Primary: demoivre1722 [dM22], stirling1753 [Sti53, proposition 15], euler1741 [Eul41], euler1750 [Eul50], laplace1814 [Lap14], bell1923 [Bel23], niven1969 [Niv69], henrici1974 [Hen74, chapter 1], zave1976 [Zav76], spiess1990 [Spi90], cauchy1826 [Cau26, pp. 95–113], polya1956 [Pól56], graham1994 [GKP94, chatper 7], wilf2006 [Wil06], polya1954 [Pól54, chapter 6], newton1707 [New07], struik1969 [Str69, pp. 94–95], girard1629 [Gir29], gauss1813 [Gau13, § 33, Eq. [75]], abel1826b [Abe26b, pp. 314–315], knopp1956 [Kno56], carlitz1965 [Car65], egorychev1977 [Ego77], knuth1994 [Knu94]. Secondary: henrici1977 [Hen77], henrici1986 [Hen86], knopp1951 [Kno51, section 66], polya1945 [Pól45], richter1949 [Ric49].
[Abe26a] N. H. Abel. Beweis eines ausdruckes, von welchem die binomial-formel ein einzelner fall ist. Journal für die reine und angewandte Mathematik, 1:159–160, 1826. [Crelle’s Journal].
[Abe26b] N. H. Abel. Untersuchungen über die reihe: u.s.w. Journal für die reine und angewandte Mathematik, 1:311–339, 1826. [Crelle’s Journal].
[Ald02] K. Alder. The Measure of All Things: The Seven-Year Odyssey and Hidden Error That Transformed the World. Free Press, New York, 2002.
[Apo74] T. M. Apostol. Mathematical Analysis. Addison-Wesley, Reading, Mass., 2nd edition, 1974.
[Arb00] L. F. A. Arbogast. Du calcul des Dérivations. Landmarks of science. Levrault, Strasbourg, 1800.
[BDM60] D. E. Barton, F. N. David, and M. Merrington. Tables for the solution of the exponential equation . Biometrika, 47:439–445, 1960.
[BDM63] D. E. Barton, F. N. David, and M. Merrington. Tables for the solution of the exponential equation . Biometrika, 50:169–176, 1963.
[Bel23] E. T. Bell. Euler algebra. Transactions of the American Mathematical Society, 25:135–154, 1923.
[Bel90] E. T. Bell. The Last Problem. Spectrum. Mathematical Association of America, [Washington, D.C.], revised and updated by underwood dudley edition, 1990.
[Ber28] D. Bernoulli. Observationes de seriebus. Commentarii academiae scientiarum imperialis Petropolitanae, 3:85–100, 1728.
[Ber57] G. M. Bergman. A number system with an irrational base. Mathematics Magazine, 31:98–110, 1957.
[Bin13] J. Binet. Mémoire sur un systeme de formules analytiques, et leur application a des considerations géométriques. Journal de l’Ecole Polytechnique, 9:280–354, 1813.
[Bon57] B. Boncampagni. Scritti di Leonardo Pisano, volume 1. Tipografia delle Scienze Matematiche e Fisiche, Rome, 1857.
[Bon62] B. Boncampagni. Scritti di Leonardo Pisano, volume 2. Tipografia delle Scienze Matematiche e Fisiche, Rome, 1862.
[Bus09] E. Busche. Zur theorie der funktionen. Journal für die reine und angewandte Mathematik, 136:42, 1909. [Crelle’s Journal].
[Bus17] W. H. Bussey. The origin of mathematical induction. American Mathematical Monthly, 24:199–207, 1917.
[Caj13a] F. Cajori. History of the exponential and logarithmic concepts. American Mathematical Monthly, 20:5–14, 1913.
[Caj13b] F. Cajori. History of the exponential and logarithmic concepts. American Mathematical Monthly, 20:35–47, 1913.
[Caj13c] F. Cajori. History of the exponential and logarithmic concepts. American Mathematical Monthly, 20:75–84, 1913.
[Caj13d] F. Cajori. History of the exponential and logarithmic concepts. American Mathematical Monthly, 20:107–117, 1913.
[Caj13e] F. Cajori. History of the exponential and logarithmic concepts. American Mathematical Monthly, 20:148–151, 1913.
[Caj13f] F. Cajori. History of the exponential and logarithmic concepts. American Mathematical Monthly, 20:173–182, 1913.
[Caj13g] F. Cajori. History of the exponential and logarithmic concepts. American Mathematical Monthly, 20:205–210, 1913.
[Caj18] F. Cajori. Origin of the name“mathematical induction”. American Mathematical Monthly, 25:197–201, 1918.
[Cap93] A. Capelli. L’analisi algebrica e l’inerpretazione fattoriale delle potenze. Giornale di Matematiche di Battaglini, 31:291–313, 1893.
[Car65] L. Carlitz. Multiple sums and generating functions. Collectanea Mathematica, 17:281–296, 1965.
[Cau15] A. L. Cauchy. Mémoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et de signes contraires par suite des transpositions opérées entre les variables qu’elles renferment. Journal de l’Ecole Polytechnique, 10:29–112, 1815.
[Cau26] A. L. Cauchy. Exercices de Mathematique, volume 1. 1826.
[Cau40] A. L. Cauchy. Exercices d’analyse et de physique mathematique, volume 1. Bachelier, Paris, 1840.
[Cau41] A. L. Cauchy. Exercices d’analyse et de physique mathematique, volume 2. Bachelier, Paris, 1841.
[Cau43] A. L. Cauchy. Analyse mathématique. Comptes rendus de l’Académie des sciences, 17:523–531, 1843.
[Cau44] A. L. Cauchy. Exercices d’analyse et de physique mathematique, volume 3. Bachelier, Paris, 1844.
[Cau47] A. L. Cauchy. Exercices d’analyse et de physique mathematique, volume 4. Bachelier, Paris, 1847.
[CG96] J. H. Conway and R. K. Guy. The Book of Numbers. Copernicus, New York, NY, 1996.
[Cha32] G. Chakravarti. Growth and development of permutations and combinations in india. Bulletin of Calcutta Mathematical Society, 24:79–88, 1932.
[Che82] P. L. Chebyshev. O priblizhennyh vyrazhenijah odnih intergralov cherez drugie. Soobschenija i Protokoly Zasedanij Mathematicheskogo Obschestva pri Imperatorskom Khar’kovskom Universitete, 2:93–98, 1882. [Reprinted in [Che07, pp. 716–719]].
[Che07] P. L. Chebyshev. Oeuvres, volume 2. Commissionaires de l’Académie impériale des sciences, St.-Pétersbourg, 1907.
[Che72] T. C. Chen. Automatic computation of exponentials, logarithms, ratios and square roots. IBM Journal of Research and Development, 16:380–388, 1972.
[Chu85] A. Church. The Calculi of Lambda Conversion. Annals of Mathematical Studies. Princeton University Press, Princeton, 1985.
[Chu96] A. Church. Introduction to Mathematical Logic. Princeton landmarks in mathematics and physics. Princeton University Press, Princeton, N.J., 1996.
[Chu19] A. Church. The Collected Works of Alonzo Church. MIT Press, Cambridge, 2019.
[CK95] C. Cooper and R. E. Kennedy. Proof of a result by jarden by generalizing a proof by carlitz. Fibonacci Quarterly, 33:304–311, 1995.
[Con01] J. H. Conway. On Numbers and Games. A. K. Peters, Natik, Mass., 2nd edition, 2001.
[Coo79] T. A. Cook. The Curves of Life. Dover Publications, New York, 1979.
[Cop04] B. J. Copeland. The Essential Turing: Seminal Writings in Computing, Logic, Philosophy, Artificial Intelligence, and Artificial Life plus The Secrets of Enigma. Oxford University Press, New York, 2004.
[Cox53] H. S. M. Coxeter. The golden section, phyllotaxis, and wythoff’s game. Scripta Mathematica, 19:135–143, 1953.
[CPS13] B. J. Copeland, C. J. Posy, and Oron Shagrir. Computability: Turing, Gödel, Church, and Beyond. MIT Press, Cambridge, 2013.
[Cul07] C. Cullen. The suán shú shū ..., “writings on reckoning”: Rewriting the history of early chinese mathematics in the light of an excavated manuscript. Historia Mathematica, 34:10–44, 2007.
[Dav59] P. J. Davis. Leonhard euler’s integral: A historical profile of the gamma function. American Mathematical Monthly, 66:849–869, 1959.
[Dav04] M. Davis. The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions. Dover Publications, Mineola, NY, 2004.
[dB57] F. Faá di Bruno. Note sur une nouvelle formule de calcul différentiel. Quarterly Journal of Pure and Applied Mathematics, 1:359–360, 1857.
[Dic02] L. E. Dickson. Theorems on the residues of multinomial coefficients with respect to a prime modulus. Quarterly Journal of Pure and Applied Mathematics, 33:383–384, 1902.
[Dic05a] L. E. Dickson. History of the Theory of Numbers, volume 1. Dover Publications, New York, 2005.
[Dic05b] L. E. Dickson. History of the Theory of Numbers, volume 2. Dover Publications, New York, 2005.
[Dic05c] L. E. Dickson. History of the Theory of Numbers, volume 3. Dover Publications, New York, 2005.
[Dij75] E. W. Dijkstra. Guarded commands, nondeterminacy and formal derivation of programs. Communications of the ACM, 18:453–457, 1975.
[Dij76] E. W. Dijkstra. A Discipline of Programming. Prentice-Hall, Englewood Cliffs, 1976.
[Dix91] A. C. Dixon. On the sum of the cubes of the coefficients in a certain expansion by the binomial theorem. Messenger of Mathematics, 20:79–80, 1891.
[Dix03] A. C. Dixon. Summation of a certain series. Proceedings of the London Mathematical Society, 35:285–289, 1903.
[dM22] A. de Moivre. De fractionibus algebraicis radicalitate immunibus ad fractiones simpliciores reducendis, deque summandis terminis quarumdam serierum aequali intervallo a se distantibus. Philosophical Transactions of the Royal Society of London, 32:162–178, 1722.
[dM30] A. de Moivre. Miscellanea Analytica. Excudebant J. Tonson & J. Watts, London, 1730.
[dM38] A. de Moivre. The Doctrine of Chances: Or, a Method of Calculating the Probability of Events in Play. H. Woodfall, London, 2nd edition, 1738.
[Dou07] J. Dougall. On vandermonde’s theorem, and some more general expansions. Proceedings of the Edinburgh Mathematical Society, 25:114–132, 1907.
[Dut84] J. Dutka. The early history of the hypergeometric function. Archive for History of Exact Sciences, 31:15–34, 1984.
[DZ91] P. Diaconis and S. Zabell. Closed form summation for classical distributions: Variations on a theme of de moivre. Statistical Science, 6:284–302, 1991.
[Ego77] G. P. Egorychev. Integral’noe predstavlenie i vychislenie kombinatornykh summ. Nauka, Sibirskoe otd-nie, 1977. [Translated in [Ego84]].
[Ego84] G. P. Egorychev. Integral Representation and the Computation of Combinatorial Sums. American Mathematical Society, 1984. [Translated from [Ego77]].
[EH90] P. Eisele and K. P. Hadeler. Game of cards, dynamical systems, and a characterization of the floor and ceiling functions. American Mathematical Monthly, 97:475–477, 1990.
[Eis44] G. Eisenstein. Geometrischer beweis des fundamentaltheorems für die quadratischer reste. Journal für die reine und angewandte Mathematik, 28:246–248, 1844. [Crelle’s Journal].
[Euc06] Euclid. The Elements: Books I–XIII. Barnes & Noble, New York, complete and unabridged edition, 2006.
[Eul29] L. Euler. correspondence to Christian Goldbach, October 13 1729. [Reprinted in [Fus43, pp. 3–7]].
[Eul34] L. Euler. De progressionibus harmonicis observationes. Commentarii academiae scientiarum imperialis Petropolitanae, 7:150–161, 1734.
[Eul41] L. Euler. Observationes analyticae variae de combinationibus. Commentarii academiae scientiarum imperialis Petropolitanae, 13:64–93, 1741.
[Eul50] L. Euler. De partitione numerorum. Novi commentarii academiae scientiarum imperialis Petropolitanae, 3:125–169, 1750.
[Eul74] L. Euler. Demonstratio theorematis nevtoniani de evolvtione potestatum binomii pro casibus quibus exponentes non sunt sumeri integri. Novi commentarii academiae scientiarum imperialis Petropolitanae, 19:103, 1774.
[Eul24] L. Euler. Institutionum calculi integralis, volume 1. Impensis Academiae Imperialis Scientiarum, Petropoli, 3rd edition, 1824.
[Eul27a] L. Euler. Institutionum calculi integralis, volume 2. Impensis Academiae Imperialis Scientiarum, Petropoli, 3rd edition, 1827.
[Eul27b] L. Euler. Institutionum calculi integralis, volume 3. Impensis Academiae Imperialis Scientiarum, Petropoli, 3rd edition, 1827.
[Eul45] L. Euler. Institutionum calculi integralis, volume 4. Impensis Academiae Imperialis Scientiarum, Petropoli, 3rd edition, 1845.
[FB94] R. W. Floyd and R. Beigel. The Language of Machines. Computer Science Press, New York, 1994.
[Fey99] R. P. Feynman. Feynman Lectures on Computing. Perseus Books, Cambridge, Mass., 1999.
[Fin47] N. J. Fine. Binomial coefficients modulo a prime. American Mathematical Monthly, 54:589–592, 1947.
[Flo67] R. W. Floyd. Assigning meanings to programs. Proceedings of Symposia in Applied Mathematics, 19:19–32, 1967.
[Flo72] R. W. Floyd. Toward interactive design of correct programs. Information Processing 71, Proceedings of IFIP Congress 71, 1:7–10, 1972.
[Fös35] W. A. Föstemann. Einfacher beweis eines satzes der combinationslehre. Journal für die reine und angewandte Mathematik, 13:237, 1835. [Crelle’s Journal].
[Fra02] J. B. Fraleigh. A First Course in Abstract Algebra. Addison-Wesley, Boston, 7th edition, 2002.
[Fre47] R. L. Frey, editor. The New Complete Hoyle. Stamford House, New York, 1947.
[Fus43] P. H. Fuss. Correspondance mathématique et physique de quelques célébres geométres du XVIIiéme siécle, volume 1. l’Académie impériale des sciences, St. Petersbourg, 1843.
[Gar61] M. Gardner. The 2nd Scientific American Book of Mathematical Puzzles and Diversions. Simon and Schuster, New York, 1961. [Reprinted in [Gar08b]].
[Gar86] M. Gardner. Knotted Doughnuts. W. H. Freeman, New York, 1986.
[Gar08a] M. Gardner. Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi: Martin Gardner’s First Book of Mathematical Puzzles and Games. The New Martin Gardner Mathematical Library. Cambridge University Press, 2008.
[Gar08b] M. Gardner. Origami, Eleusis, and the Soma Cube: Martin Gardner’s Mathematical Diversions. The New Martin Gardner Mathematical Library. Cambridge University Press, 2008.
[Gar09] M. Gardner. Sphere Packing, Lewis Carrol and Reversi. The New Martin Gardner Mathematical Library. Cambridge University Press, 2009.
[Gar14] M. Gardner. Knots and Borromean Rings, Rep-Tiles, and Eight Queens: Martin Gardner’s Unexpected Hanging. The New Martin Gardner Mathematical Library. Cambridge University Press, 2014.
[Gau08] C. F. Gauss. Summatio quarumdam serierum singularium. Commentationes societatis regiæ scientiarum Gottingensis recentiores, 1:147–186, 1808. [Page numbers from Kraus reprint of 1970; reprinted in [Gau63, pp. 9–45]].
[Gau13] C. F. Gauss. Disquisitiones generales circa seiem infinitam pars prior. Commentationes societatis regiae scientiarum Gottingensis recentiores, 2, 1813. [Delivered 1812; reprinted in [Gau76, p. 123–163; see also unpublished sequel pp. 207–229]].
[Gau63] C. F. Gauss. Werke, volume 2. Dieterich, Göttingen, 1863.
[Gau76] C. F. Gauss. Werke, volume 3. Dieterich, Göttingen, 1876.
[Gau65] C. F. Gauss. Disquisitiones Arithmeticae. Yale University Press, New Haven, 1965.
[Gir29] A. Girard. Invention Nouvelle en Algébre. Amsterdam, 1629. [Reprinted in [Gir84]].
[Gir84] A. Girard. Invention Nouvelle en Algébre. Leiden, 1884. [Reprint of [Gir29]].
[GK66] H. W. Gould and J. Kaucký. Evaluation of a class of binomial coefficient summations. Journal of Combinatorial Theory, 1:233–247, 1966.
[GKP94] R. L. Graham, D. E. Knuth, and O. Patashnik. Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley, Reading, Mass., 2nd edition, 1994.
[Göd01a] K. Gödel. Collected Works: Volume I: Publications 1929–1936, volume 1. Oxford University Press, New York, 2001.
[Göd01b] K. Gödel. Collected Works: Volume II: Publications 1938–1974, volume 2. Oxford University Press, New York, 2001.
[Göd01c] K. Gödel. Collected Works: Volume III: Unpublished Essays and Lectures, volume 3. Oxford University Press, New York, 2001.
[Göd13a] K. Gödel. Collected Works: Volume IV: Correspondence A–G, volume 4. Oxford University Press, New York, 2013.
[Göd13b] K. Gödel. Collected Works: Volume V: Correspondence H–Z, volume 5. Oxford University Press, New York, 2013.
[Gol65] S. W. Golomb. A geometric proof of a famous identity. Mathematical Gazette, 49:198–200, 1965.
[Goo61] I. J. Good. The multivariate saddlepoint method and chi-squared for the multinomial distribution. Annals of Mathematical Statistics, 32:540–541, 1961.
[Gou56] H. W. Gould. Some generalizations of vandermonde’s convolution. American Mathematical Monthly, 63:84–91, 1956.
[Gou61] H. W. Gould. Some relations involving the finite harmonic series. American Mathematical Monthly, 34:317–321, 1961.
[GR90] G. Gasper and M. Rahman. Basic Hypergeometric Series. Cambridge University Press, New York, 1990.
[HA72] C. A. R. Hoare and D. C. S. Allison. Incomputability. Computing Surveys, 4:169–178, 1972.
[Ham05] G. Hamel. Eine basis aller zahlen und die unstetigen lösungen der funktionalgleichung: . Mathematische Annalen, 60:459–462, 1905.
[Ham97] R. W. Hamming. The Art of Doing Science and Engineering. Gordon and Breach, Australia, 1997.
[Han64] H. Hankel. Die euler’schen integrale bei unbeschränkter variabilität des argumentes. Zeitscrift für angewandte Mathematik und Physik, 9:1–21, 1864.
[Har99] G. H. Hardy. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. American Mathematial Society, Providence, R. I., 1999.
[HB97] D. K. Hathaway and S. L. Brown. Fibonacci powers and a fascinating triangle. College Mathematics Journal, 28:124–128, 1997.
[Hea08] T. L. Heath. Diophantus of Alexandria: A Study in the History of Greek Algebra. Heath Press, 2nd edition, 2008.
[Hei47] E. Heine. Untersuchungen über die reihe . Journal für die reine und angewandte Mathematik, 34:285–328, 1847. [Crelle’s Journal].
[Hen60] L. Henkin. On mathematical induction. American Mathematical Monthly, 67:323–338, 1960.
[Hen74] P. Henrici. Applied and Computational Complex Analysis, volume 1. Wiley, 1974.
[Hen77] P. Henrici. Applied and Computational Complex Analysis, volume 2. Wiley, 1977.
[Hen86] P. Henrici. Applied and Computational Complex Analysis, volume 3. Wiley, 1986.
[Her84] C. Hermite. Sur quelques conséquences arithmétiques des formules de la théorie des fonctions elliptiques. Acta Mathematica, 5:315, 1884.
[Her17] C. Hermite. Œuvres de Charles Hermite, volume 4. Académie des sciences, France, 1917.
[Hey02] A. Hey. Feynman and Computation. Perseus Books, Cambridge, Mass., 2002.
[Hoa71] C. A. R. Hoare. Proof of a program: Find. Communications of the ACM, 14:39–45, 1971.
[Hof99] D. R. Hofstadter. Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books, New York, 20th anniversary edition, 1999.
[Hui07] Y. Hui. Yongle Encyclopedia, page [unknown]. Chengzu, the Yongle Emperor, Nanjing, 1407. [Jia Xian’s Shi Suo Suan Shu; appears in [Zhu06]].
[Hur02] A. Hurwitz. Über abel’s verallgemeinerung der binomischen formel. Acta Mathematica, 26:199–203, 1902.
[HW08] G. H. Hardy and E. W. Wright. An Introduction to the Theory of Numbers. Oxford University Press, New York, 6th edition, 2008.
[Hwa95] H. K. Hwang. Asymptotic expansions for the stirling numbers of the first kind. Journal of Combinatorial Theory, A71:343–351, 1995.
[III72] J. Bernoulli III. Sur une nouvelle espece de calcul. Recueil pour les Astronomes, 1:255–284, 1772.
[Jac65] S. Jacob. Ein new und wohlgegründt Rechenbuch, auf den Linien und Ziffern, Sampt der Welschen Practica... Feyerabend & Hüter, Frankfurt, 1565.
[Jac46] C. G. J. Jacobi. Über einige der binomialreihe analoge reihen. Journal für die reine und angewandte Mathematik, 32:197–204, 1846. [Crelle’s Journal].
[Jar66] D. Jarden. Recurring Sequences. Riveon Lematematika, Jerusalem, 2nd edition, 1966.
[JM49] D. Jarden and T. Motzkin. The product of sequences with a common linear recursion formula of order 2. Riveon Lematematika, 3:25–27,38, 1949. [Translated in [HB97]].
[Kar35] J. Karamata. Theéorémes sur la sommabilité exponentielle et d’autres sommabilités s’y rattachant. Mathematica (Cluj), pages 164–178, 1935.
[Kep08] J. Kepler. correspondence to Joachim Tanckius, May 12 1608. [Reprinted in [Kep55, pp. 154–165,429]].
[Kep55] J. Kepler. Gesammelte Werke, volume 16. C. H. Beck, Munich, 1955.
[Kep14] J. Kepler. The Six-Cornered Snowflake. Oxford Classic Texts in the Physical Sciences. Oxford University Press, Oxford, 2014.
[Kin72] J. C. King. A program verifier. Information Processing 71, Proceedings of IFIP Congress 71, 1:234–249, 1972.
[KKWvL62] M. S. Klamkin, W. H. M. Kantor, J. W. Wrench Jr., and J. H. van Lint. 4946. American Mathematical Monthly, 69:239–240, 1962.
[Kla68] D. A. Klarner. Partitions of into distinct fibonacci numbers. Fibonacci Quarterly, 6:235–244, 1968.
[Kle12] S. C. Kleene. Introduction to Metamathematics. Literary Licensing, Whitefish, MT, 2012.
[Kno51] K. Knopp. Theory and Application of Infinite Series. Blackie, London, 2nd edition, 1951.
[Kno56] K. Knopp. Infinite Sequences and Series. Dover, 1956.
[Knu63] D. E. Knuth. Computer-drawn flowcharts. Communications of the ACM, 6:555–563, 1963.
[Knu71] D. E. Knuth. Subspaces, subsets, and partitions. Journal of Combinatorial Theory, A10:178–180, 1971.
[Knu86a] D. E. Knuth. Computer Modern Typefaces, volume E. Addison-Wesley, Reading, Mass., 1986.
[Knu86b] D. E. Knuth. The Metafont Book, volume C. Addison-Wesley, Reading, Mass., 1986.
[Knu86c] D. E. Knuth. Metafont: The Program, volume D. Addison-Wesley, Reading, Mass., 1986.
[Knu86d] D. E. Knuth. TeX: The Program, volume B. Addison-Wesley, Reading, Mass., 1986.
[Knu86e] D. E. Knuth. The TeXbook, volume A. Addison-Wesley, Reading, Mass., 1986.
[Knu92] D. E. Knuth. Two notes on notation. American Mathematical Monthly, 99:403–422, 1992.
[Knu94] D. E. Knuth. Bracket notation for the ”coefficient of” operator. In A. W. Roscoe, editor, A Classical Mind: Essays in Honour of C. A. R. Hoare, pages 247–258. Prentice-Hall, 1994.
[Kra01] C. Kramp. Élémens d’Arithmétique. Chez Oedenkoven, Cologne, 1801.
[Kra08] C. Kramp. Élémens d’Arithmétique Universelle. De l’imprimerie de Th. F. Thiriart, Cologne, 1808.
[Kra88] C. Krattenthaler. Operator methods and lagrange inversion: A unified approach to lagrange formulas. Transactions of the American Mathematical Society, 305:431–465, 1988.
[Kra90] C. Krattenthaler. Generating functions for plane partitions of a given shape. Manuscripta Mathematica, 69:177–178, 1990.
[Kum52] E. E. Kummer. Über die ergänzungssätze zu den allgemeinen reciprocitätsgesetzen. Journal für die reine und angewandte Mathematik, 44:93–146, 1852. [Crelle’s Journal].
[KW89] D. E. Knuth and H. S. Wilf. The power of a prime that divides a generalized binomial coefficient. Journal für die reine und angewandte Mathematik, 396:212–219, 1989. [Crelle’s Journal].
[Lap14] P. S. Laplace. Théorie Analytique des Probabilités. Paris, 2nd edition, 1814.
[Leg08] A. M. Legendre. Essai sur la Théorie des Nombres. Chez Courcier, Paris, 2nd edition, 1808.
[Lég37] E. Léger. Note sur le partage d’une droite en moyenne et extrême, et sur un probléme d’arithmétique. Correspondance mathématique et physique, 9:483–485, 1837.
[Lek51] C. G. Lekkerkerker. Voorstelling van natuurlijke getallen door een som van getallen van fibonacci. Technical Report ZW 30/51, Stichting Mathematisch Centrum, 1951.
[Lek52] C. G. Lekkerkerker. Voorstelling van natuurlijke getallen door een som van getallen van fibonacci. Simon Stevin, 29:190–195, 1952. [Originally in [Lek51]].
[LeV56] W. J. LeVeque. Topics in Number Theory, volume 2. Addison-Wesley, Reading, Mass., 1956.
[Lin57] V. S. Linskii. Calculation of elementary functions on automatic digital machines. Vychislitel’naia matematika, 2:90–119, 1957.
[Liv03] M. Livio. The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number. Broadway Books, New York, 2003.
[Log87] B. F. Logan. Polynomials related to the stirling numbers. Technical Memorandum 11218-870810-23TM, AT&T Bell Labs, August 10, 1987.
[Lov93] L. Lovász. Combinatorial Problems and Exercises. North-Holland, Amsterdam, 1993.
[Luc76a] É. Lucas. Questions proposées. Nouvelle correspondance mathématique, 2:94–96, 1876.
[Luc76b] É. Lucas. Sur les rapports qui existent entre la théorie des nombres et le calcul intégral. Comptes Rendus hebdomadaires des séances de l’Académie des Sciences, 82:1303–1305, 1876.
[Luc78a] É. Lucas. Théorie des fonctions numériques simplement périodiques [part 1 of 3]. American Journal of Mathematics, 1:184–196, 1878.
[Luc78b] É. Lucas. Théorie des fonctions numériques simplement périodiques [part 2 of 3]. American Journal of Mathematics, 1:197–240, 1878.
[Luc78c] É. Lucas. Théorie des fonctions numériques simplement périodiques [part 3 of 3]. American Journal of Mathematics, 1:289–321, 1878.
[Mac02] P. A. MacMahon. The sums of powers of the binomial coefficients. Quarterly Journal of Pure and Applied Mathematics, 33:274–288, 1902.
[Mah12] Mahāvīra. The Ganita-sāra-sangraha of Mahāvīrācārya, with English translation and notes by M. Raṅgācārya. Government Press, Madras, 1912.
[Mar82] A. A. Markov. Sur une question de jean bernoulli. Mathematische Annalen, 19:27–36, 1882.
[Mar54] A. A. Markov. The Theory of Algorithms. Works of the Mathematical Institute Imeni V. A. Steklov. Academy of Sciences of the USSR, Moscow, 1954. [Later revised and enlarged in [MN88]].
[Mar92] G. Markowsky. Misconceptions about the golden ratio. College Mathematics Journal, 23:2–19, 1992.
[Mat93] Y. Matiyasevich. Hilbert’s 10th Problem. MIT Press, Cambridge, 1993.
[Meg62] J. E. Meggitt. Pseudo division and pseudo multiplication processes. IBM Journal of Research and Development, 6:210–226, 1962.
[Meg63] J. E. Meggitt. Digit-by-digit methods for polynomials. IBM Journal of Research and Development, 7:237–245, 1963.
[MJ84] F. L. Morris and C. B. Jones. An early program proof by alan turing. Annals of the History of Computing, 6:139–143, 1984.
[MN88] A. A. Markov and N. M. Nagorny. The Theory of Algorithms. Kluwer, Dordrecht, 1988.
[Mor58] L. J. Mordell. Integral formulae of arithmetical character. Journal of the London Mathematical Society, 33:371–375, 1958.
[MW58a] L. Moser and M. Wyman. Asymptotic development of the stirling numbers of the first kind. Journal of the London Mathematical Society, 33:133–146, 1958.
[MW58b] L. Moser and M. Wyman. Stirling numbers of the second kind. Duke Mathemtical Journal, 25:29–43, 1958.
[Nan63] T. S. Nanjundiah. 5022. American Mathematical Monthly, 70:575–577, 1963.
[Nau66] P. Naur. Proof of algorithms by general snapshots. BIT, 6:310–316, 1966.
[Nee54] J. Needham. Science and Civilisation in China, volume 1. Cambridge University Press, Cambridge, 1954.
[Nee56] J. Needham. Science and Civilisation in China, volume 2. Cambridge University Press, Cambridge, 1956.
[Nee59] J. Needham. Science and Civilisation in China, volume 3. Cambridge University Press, Cambridge, 1959.
[New76a] I. Newton. correspondence to Oldenburg, June 13 1676. [Reprinted in [Str69, pp. 284–291]].
[New76b] I. Newton. correspondence to Oldenburg, October 24 1676. [Reprinted in [Str69, pp. 284–291]].
[New07] I. Newton. Arithmetica Universalis. 1707. [Translated in [New20]].
[New20] I. Newton. Universal Arithmetick. 1720. [Translated from [New07]].
[Niv69] I. Niven. Formal power series. American Mathematical Monthly, 76:871–889, 1969.
[Pas65] B. Pascal. Traité du Triangle Arithmétique. G. Desprez, Paris, 1665.
[Pas87] E. Pascal. Sopra una formula numerica. Giornale di Matematiche di Battaglini, 25:45–49, 1887.
[Pfa97] J. F. Pfaff. Observationes analyticae. Nova Acta Academiae Scientiarum Petropolitanae, 11:37–57, 1797.
[Piṅ74] Piṅgala. The Chhandah Sútra of Pingala Áchárya with the Commentary of Haláyudha. Ganeśa, Calcutta, 1874.
[Pis02] L. Pisano. Liber Abaci. [Unpublished manuscript], 1202. [Reprinted in [Bon57], [Bon62]; translated in [Pis03]].
[Pis03] L. Pisano. Fibonacci’s Liber Abaci: A Translation into Modern English of Leonardo Pisano’s Book of Calculation. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, New York, 2003.
[PNS06] K. Patwardhan, S. A. Naimpally, and S. L. Singh. Līlāvatī of Bhāskarācācārya: A Treatise of Mathematics of Vedic Tradition. Motilal Banarsidass, Delhi, reprint edition, 2006.
[Poi96] H. Poincaré. Calcul des Probabilités. Carré, Paris, 1896.
[Pól45] G. Pólya. How to Solve It: A New Aspect of Mathematical Method. Princeton University Press, Princeton, N.J., 1945.
[Pól54] G. Pólya. Induction and Analogy in Mathematics. Princeton University Press, Princeton, N.J., 1954.
[Pól56] G. Pólya. On picture writing. American Mathematical Monthly, 63:689–697, 1956.
[PWZ96] M. Petkovšek, H. S. Wilf, and D. Zeilberger. . A. K. Peters, Wellesley, Mass., 1996.
[Ram34] C. Ramus. Solution générale d’un probléme d’analyse combinatoire. Journal für die reine und angewandte Mathematik, 11:353–355, 1834. [Crelle’s Journal].
[Ram77] L. Ramshaw. Binomial coefficients with non-integral lower index. Information Processing Letters, 6:223–226, 1977.
[Ras72] R. Rashed. L’induction mathématique al-karajī, as-samaw’al. Archive for History of Exact Sciences, 9:1–21, 1972.
[Red96] R. M. Redheffer. Use of a differential inequality in combinatorics. American Mathematical Monthly, 103:62–63, 1996.
[Rib79] P. Ribenboim. 13 Lectures on Fermat’s Last Theorem. Springer-Verlag, New York, 1979.
[Ric49] H. Richter. Ein einfacher beweis der newtonschen und der waringschen formel für die potenzsummen. Archiv der Mathematik, 2:1–4, 1949.
[Rio62] J. Riordan. Generaring functions for powers of fibonacci numbers. Duke Mathematical Journal, 29:5–12, 1962.
[Rog95] L. J. Rogers. Third memoir on the expansion of certain infinite products. Proceedings of the London Mathematical Society, 26:15–32, 1895.
[Rot93] H. A. Rothe. Formulæ de Serierum Reversione. Sommer, Lipsiae, 1793.
[Rot11] H. A. Rothe. Systematisches Lehrbuch der Arithmetik. Barth, Leipzig, 1811.
[Sch20] F. Schweins. Analysis. Mohr und Winter, Heidelberg, 1820.
[Sch28] H. F. Scherk. Aufgaben und lehrsätze, erstere aufzulösen, letztere zu beweisen. Journal für die reine und angewandte Mathematik, 3:97, 1828. [Crelle’s Journal].
[Sch70] A. J. Schwenk. Take-away games. Fibonacci Quarterly, 8:225–234, 1970.
[Sch95] P. Schreiber. A supplement to j. shallit’s paper ’origins of the analysis of the euclidean algorithm’. Historia Mathematica, 22:422–424, 1995.
[Sha94] J. Shallit. Origins of the analysis of the euclidean algorithm. Historia Mathematica, 21:401–419, 1994.
[Sin85] P. Singh. The so-called fibonacci numbers in ancient and medieval india. Historia Mathematica, 12:229–244, 1985.
[Sin98] S. Singh. Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem. Anchor, New York, 1998.
[SMG62] D. L. Shell, Y. Matsuoka, and H. W. Gould. 4960. American Mathematical Monthly, 69:571–572, 1962.
[Smi58a] D. E. Smith. History of Mathematics, volume 1. Dover, New York, 1958.
[Smi58b] D. E. Smith. History of Mathematics, volume 2. Dover, New York, 1958.
[Soc38] Society for the Diffusion of Useful Knowledge. Penny Cyclopædia, volume 12. Knight, London, 1838.
[Spi90] J. Spieß. Some identities involving harmonic numbers. Mathematics of Computation, 55:839–863, 1990.
[SS60] J. L. Selfridge and D. L. Silverman. E1408. American Mathematical Monthly, 67:924–925, 1960.
[Sti53] J. Stirling. Methodus Differentialis. Ric. Manby, London, 1753. [Translated in [Twe03]].
[Sti90] L. Stickelberger. Ueber eine verallgemeinerung der kreistheilung. Mathematische Annalen, 37:321–367, 1890.
[Sto76] K. B. Stolarsky. Beatty sequences, continued fractions, and certain shift operators. Canadian Mathematical Bulletin, 19:473–482, 1976.
[Str69] D. J. Struik. Source Book in Mathematics, 1200-1800. Harvard University Press, Cambridge, Mass., 1969.
[SW62] D. Shanks and J. W. Wrench Jr. Calculation of to 100,000 decimals. Mathematics of Computation, 16:76–99, 1962.
[Syl57] J. J. Sylvester. On the partition of numbers. Quarterly Journal of Pure and Applied Mathematics, 1:141–152, 1857.
[Tem93] N. M. Temme. Asymptotic estimates of stirling numbers. Studies in Applied Mathematics, 89:233–243, 1993.
[Tod61] J. Todd. Computational problems concerning the hilbert matrix. Journal of Research of the National Bureau of Standards, 65:19–22, 1961.
[Tor95] G. Torelli. Qualche formola relativa all’inerpretazione fattoriale delle potenze. Giornale di Matematiche di Battaglini, 33:179–182, 1895.
[Tos39] L. Toscano. Numeri di stirling generalizzati, operatori differenziali e polinomi ipergeometrici. Commentationes Pontificia Academica Scientarum, 3:721–757, 1939.
[Tur49] A. M. Turing. Report of a Conference on High Speed Automatic Calculating Machines, 22–25 June 1949, pages 67–68. Cambridge University, Cambridge, 1949. [Reprinted with commentary in [MJ84]].
[Twe03] I. Tweddle. James Stirling’s Method Differentialis: An Annotated Translation of Stirling’s Text. Springer-Verlag, New York, 2003.
[Uhl55] H. S. Uhler. Exact values of 995! and 1000!, with skeleton tables of antecedent constants. Scripta Mathematica, 21:266–267, 1955.
[Vac99] G. Vacca. Sui manoscritti inediti di leibniz. Bollettino di bibliografia e storia delle scienze matematiche, 2:113–116, 1899.
[Van72] A. Vandermonde. Memoire sur des irrationnelles de différens ordres avec une application au cercle. Mémoires de l’Académie Roayle des Sciences, 1:489–498, 1772.
[vE26] A. von Ettingshausen. Die Combinatorische Analysis. J. B. Wallishausser, Wien, 1826.
[vH02] J. van Heijenoort. From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Harvard University Press, Cambridge, 2002.
[vN63] J. von Neumann. Collected Works, volume 5. Pergamon Press, New York, 1963.
[Vor92] N. N. Vorob’ev. Chisla Fibonachchi. Nauka, Moscow, 6th edition, 1992.
[Vor02] N. N. Vorob’ev. Fibonacci Numbers. Birkhauser Verlag, Boston, 2002. [Translated from [Vor92]].
[Wal38] H. S. Wall. On the th derivative of . Bulletin of the American Mathematical Society, 44:395–398, 1938.
[War79] E. Waring. Problems concerning interpolations. Philosophical Transactions of the Royal Society of London, 69:64–67, 1779.
[War82] E. Waring. Mediationes Algebraicæ. J. Archdeacon, Cambridge, 3rd edition, 1782. [Translated in [Wee91]].
[Wee91] D. Weeks. Meditationes Algebraicæ: An English Translation of the Work of Edward Waring. American Mathematical Society, Providence, R.I., 1991.
[Whi63] M. J. Whinihan. Fibonacci nim. Fibonacci Quarterly, 1:9–12, 1963.
[Wil93] H. S. Wilf. The asymptotic behavior of the stirling numbers of the first kind. Journal of Combinatorial Theory, A64:344–349, 1993.
[Wil95] A. Wiles. Modular elliptic curves and fermat’s last theorem. Annals of Mathematics, 141:443–551, 1995.
[Wil06] H. S. Wilf. generatingfunctionology. Academic Press, 3rd edition, 2006.
[WR11] A. N. Whitehead and B. Russell. Principia Mathematica, volume 1–3. Rough Draft Printing, S.l., 2011.
[Yod75] M. F. Yoder. Continuous replicative functions. Æquationes Mathematicæ, 13:251–261, 1975.
[Zav76] D. A. Zave. A series expansion involving the harmonic numbers. Information Processing Letters, 5:75–77, 1976.
[Zec72] E. Zeckendorf. Representation des nombres naturels par une somme de nombres de fibonacci ou de nombres de lucas. Bulletin de la Société Royale des Sciences de Liége, 41:179–182, 1972.
[Zem81] H. Zemanek. Algorithms in modern mathematics and computer science: Proceedings, urgench, uzbek ssr september 16-22, 1979. Lecture Notes in Computer Science, 122:1–81, 1981.
[Zhu06] S. Zhu. Si yuan yü jian [Jade Mirror of the Four Unknowns], volume 1–2. Liaoning, Shenyang, 2006.