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Any assistance in providing additional information for the following items would be greatly appreciated. In many cases, the missing portions are article names, volume numbers, page ranges or ending pages, publication city, etc. Where possible, the needed information is indicated explicitly, e.g., in the form [VOLUME?]. Note that in some cases, the information missing may not be explicitly indicated, but may instead simply be a copy of a citation in its current form.

A typical journal citation should be of the form

        Author, Initials. "Title." Journal Volume, [Beginning Page]-[Ending Page], Year.

A typical book citation should be of the form

        Author, Initials. Title. City: Publisher, [Page Range], Year.

In other cases, missing information may be verifying an equation, adding a figure, etc. In such cases, the following may contain only a cryptic short-hand note from the author to himself. Over time, these cryptic notes will be replaced with more easily decipherable information requests. In the meantime, please send information to the MathWorld team.

Abelian Differential (any reference)

Alexander's Horned Sphere (Albers, D. J. Illustration accompanying "The Game of 'Life.' " Math Horizons, p. 9, Spring 1994)

Amicable Pair (Paganini, B. N. I. [TITLE?] Atti della R. Accad. Sc. Torino 2, 362 [ENDING PAGE?], 1866-1867.)

Anomalous Cancellation (Moessner, A. Scripta Math. 19. [cited in Wells] Moessner, A. Scripta Math. 20. [cited in Wells])

Antipode (add figure)

Augmentation (Graziotti, U. Polyhedra, the Realm of Geometric Beauty. San Francisco, CA: 1962.)

Ax-Kochen Isomorphism Theorem (nonprincipal maximal filter)

Baguenaudier (Kauffman, L. H. "Tangle Complexity and the Topology of the Chinese Rings." In Mathematical Approaches to Biomolecular Structure and Dynamics. New York: Springer-Verlag, pp. 1-10, 1996)

Bang's Theorem (Bang, A. S. [TITLE?] Tidskrift f. Math., [VOL?], p. 48 [PAGE RANGE], 1897. Gehrke. [INITIALS?] [TITLE?] Tidskrift for Math. [VOL?], p. 84 [PAGE RANGE], 1897. In Bang's (1951) theorem, what is w in w(K)?)

Bauer's Identical Congruence (Bauer. Nouvelles annales 2, 256-264, 1902.; Hardy, G. H. and Wright, E. M. J. London Math. Soc. 9, 38-41 and 240, 1934.;)

BBP-Type Formula (Borwein, J. M. and Chamberland, M. "A Golden Example." Unpublished manuscript.; Cloitre, B. "A BBP Formula for pi^2 in Golden Base." Unpublished manuscript.)

Benham's Wheel (Benham 1894; Brücke 1864; Finnegan and Moore 1895; Helmholtz 1856; Kunsemüller, J. Meyers Lexikon der Technik und der exakten Naturwissenschaften, Bd. 1. A - E. [CITY?] Fischer Taschenbuch Verlag, [PAGES?] 1970)

Bialtitude (add figure)

Bicentric Quadrilateral (Davis, M. A. Educ. Times 32.)

Binomial Sums (Almkvist, G.; Krattenthaler, C.; and Petersson, J. "Some New Formulas for Pi." Manuscript, 2001.; sloane: 2/3, 4/3, 10/3, 32/3, [need more terms]; sloane: 2/9, 10/27, 74/81, [need more terms]; sloane: 2/25, 81/625, 561/3125, [need more terms]; sloane: -6/125, -18/3125, 42/15625, [need more terms]; sloane: 11/250, 79/3125, 673/31250, [need more terms];)

Binormal Vector (add figure)

Bôcher's Theorem (Bôcher. "The Location of Critical Points." Amer. Math. Soc. Colloq. Publ. 34, 1950.)

Braikenridge-Maclaurin Construction (add figure)

Breeder (make sense of last sentence)

Bridge Knot (Complete number of bridge knots of 5-9 crossings)

Bridge Number (make sure I mean 05-002 here; was 05-05)

Burridge-Knopoff Model (Burridge, R. and Knopoff, L. Bull. Seis. Soc. Amer. 57, 341, 1967.; de Sousa Vieira, M. Phys. Rev. Lett. 82, 201, 1999.; Ferguson, C. D.; Klein, W.; and Rundle., J. B. Computers Physics 12, 34 1998.; Hähner, P. and Drossinos, Y. Physica A 260, 391, 1998.; Place, D. and Villedieu, P. J. Comput. Phys. 150, 332,1999.; Xu, H.-J and Knopoff, L. Phys. Rev. E 50, 3577, 1994.;)

Cage Graph (sloane: 6,10,30,42 sloane: 7,12,40,62 sloane: 8,14,50,90)

Carmichael's Conjecture (Cipra reference)

Casey's Theorem (more figures)

Cassini Ovals (Gosper says it is only the critical case where the ovals unite)

Catalan's Constant (sloane: 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, Bresse (1867) [from https://functions.wolfram.com/Constants/I/introductions/ClassicalConstants/01/])

Character Table (finish tables)

Chebyshev (Clenshaw, C. W. Mathematical Tables, Vol. 5: Chebyshev Series for Mathematical Functions. Department of Scientific and Industrial Research.)

Chess (Dawson, T. R. "A Surprise Correction." The Fairy Chess Review 6, 44, 1946; Cunningham 1889; Fabel, K. Die Schwalbe 89, 284-285, 1935.; check: Peterson 10.120)

Chevalley Groups (add correct restrictions on parameters n and q)

Chinese Hypothesis (Jeans, J. H. Messenger Math. 27, 1897-98.; Needham, J. Ch. 19 in Science and Civilisation in China, Vol. 3.)

Chió Pivotal Condensation (Hermite 1859)

Circle Covering by Arcs (Whitworth, W. A. DCC Exercises in Choice and Chance. 1897.)

Circle Power (Clifford. Collected Mathematical Papers. 1882.)

Circles-and-Squares Fractal (any reference)

Circumcevian Triangle (Kimberling, C. "Transfigured Triangle Geometry." Preprint. Mar. 5, 2005.)

Cissoid (add figure)

Complex Matrix (Hwang, C. R. "A Brief Survey on the Spectral Radius and the Spectral Distribution of Large Random Matrices with i.i.d. Entries." In Random Matrices and Their Applications. Providence, RI: Amer. Math. Soc., pp. 145-152, 1986)

Condensation (Macmillan, R. H. A New Method for the Numerical Evaluation of Determinants." J. Roy. Aeronaut. Soc. 59, 772, 1955.)

Cone Net (Nowakowski, M. Matematyka, No. 1 (29), 1954.)

Confluent Hypergeometric Differential Equation (Previato: discuss links with differential equations, geometry, and 19th century mathematics. A suggestion of where hypergeometric functions arise, a glimpse of Riemann surfaces?)

Confluent Hypergeometric Function of the First Kind (Previato: discuss links with differential equations, geometry, and 19th century mathematics. A suggestion of where hypergeometric functions arise, a glimpse of Riemann surfaces?)

Confluent Hypergeometric Function of the Second Kind (Previato: discuss links with differential equations, geometry, and 19th century mathematics. A suggestion of where hypergeometric functions arise, a glimpse of Riemann surfaces?)

Conformal Projection (Germain 1865)

Congruence (Schroeppel: for prime moduli, sqrt a = +- sqrt b, and kth roots work when k is relatively prime to p-1.)

Conic Section (Chasles, M. Traité des sections coniques. Paris, France, 1865.)

Connex (Clebsch, A. "Ueber ein neues Grundgebilde der analytischen Geometri der Ebene." Göttinger Nachr. No. 22, 1872. Reprinted in Math. Ann. 6, 203-225, 1873.; Godt. Ueber den Connex ester Ordnung und zweiter Classe. Göttingen, Germany, 1873.; Hirst. Proc. London Math. Soc. 63, 1874; Sintsof. Theory of the Connex in Space, in Connection with the Theory of Partial Differential Equations of the First Order. Publications of Kasan University, 1895.; Sintsof. "Theorie des connexes dans l'espace." Bull. des Sci. Math., p. 176, 1898. Stephanos. Bull. Sci. Math. 4, 1880.;)

Contractible (add Margherita's images contractible1.bmp, etc.)

Convex Polyhedron (Proc. CUPM Geometry Conf. Math. Assoc. Amer, No. 16, p. 21, 1967)

Correlation Dimension (fractal Whitney embedding prevalence theorem)

Cosymmedian Triangles (figure)

Cousin Primes (sloane: 1.1970449)

Császár Polyhedron (can't render properly in LiveGraphics3D)

Cubic Number (Moret Blanc 1879; Sphinx. pp. 72 and 87 1937.;)

Cyclidic Coordinates (Wangerin. Preisschriften der Jablanowski'schen Gesellschaft, No. 18, 1875-1876. Wangerin. Crelle's J. 82, 1875-1876. Wangerin. Berliner Monatsber. 1878.)

Cyclologic Center ([FINAL PAGES? DOI OR PUBLISHER URL? for Rabinowitz and Suppa 2026])

Cyclologic Triangles ([FINAL PAGES? DOI OR PUBLISHER URL? for Rabinowitz and Suppa 2026])

Daisy (Previato: Is there a place where the mathematical formulas for how many blobs one gets for given prescription (equation of the spiral, which can be algebraic or transcendental, radius of the blobs etc.)?)

Difference of Successes (check formulas; find reference)

Differentiable (Hermite 1893)

Diophantine Equation--4th Powers (Martin 1910; Baudran 1885; Euler, L. Nova Acta Acad. Petrop. as annos 1795-1796 13, 45, 1802.; Fauquembergue, E. L'intermédiaire des Math. 5, 33, 1898.; Ferrari, F. L'intermédiaire des Math. 20, 105-106, 1913.; Gérardin, A. Assoc. Franc. 39, 236, 1911. [GARBLED CITATION?]; Gérardin, A. Bull. Soc. Philomathique 3, 236, 1911. [WRONG VOLUME]; Werebrusow, A. L'intermédiaire Math. 20, 105-106, 1913. [NO SUCH AUTHOR OR ARTICLE WITH THIS CITATION])

Diophantine Equation--5th Powers (LEO can't find as cited; will provide ILL microfilm: Martin, A. Smithsonian Misc. Coll. 33, 1888.)

Diophantine Equation--8th Powers (Find original citation; Letac, A. Gazetta Mathematica 48, 68-69, 1942)

Disk Triangle Picking (VERIFY CITATION: Woolhouse, W. S. B. "Some Additional Observations on the Four-Point Problem." Mathematical Questions, with Their Solutions, from the Educational Times, Vol. 7. London, England: F. Hodgson and Son, p. 81, 1867.)

Divisor (Chalde [INITIALS?] [TITLE?] Nouv. Ann. Math. 3, 471-473, 1903.; Minin, A. P. [TITLE?] Math. Soc. Moscow 11, 632, 1883-84.; Fontené, G. [TITLE?] Nouv. Ann. Math. 2, 288, 1902.)

Dixon's Factorization Method (provide concrete example)

Donkin's Theorem (add figure)

Double-Free Set (hyphen to dash)

Doubly Periodic Function (Jacobi, C. G. J. J. für Math. 13, 55-56, 1835. Reprinted in Gesammelte Werke, Band II. pp. 25-26, 1882.)

Dowker Notation (gopher://math.berkeley.edu:70/11/Topology/VFRJones/f3to10)

Ear (add figure)

Edgeworth Series (Cramér, H. "On Some Classes of Series Used in Mathematical Statistics." Proceedings of the Sixth Scandinavian Congress of Mathematicians, Copenhagen. pp. 399-425, 1925.)

Ellipsoid Geodesic (Dragovich, V. J. Phys. A 29, L317, 1996. Dragović, V. math-ph/0008009 (2000). Knörrer, H. Invent. Math. 59, 119, 1980. Kravchenko, N. N. Vestnik Mosk. Univ Ser. 1, No. 4, 69, 1996. Tabanov, M. B. Russian Acad. Sci. Dokl. Math. 48, 438, 1994. Toth, J. A. Ann. Phys. 130, 1, 1995. Viesel, H. Archiv Math. 22, 106, 1971. Wiersig, J. and Richter, P. H. Z. Naturf. 51a, 219, 1996.)

Elongated Dodecahedron (Tutton. Crystallography and Practical Crystal Measurement. London, England, pp. 567 (Fig. 448) and 723 (Fig. 585), 1922. Fedorov. "Elemente der Gestaltenlehre." Mineralogicheskoe obshchestvo Leningrad (Verhandlungen der Russisch-Kaiserlichen Mineralogischen Gesellschaft zu St. Petersburg) 21, 1-279, 1885.)

Elongated Square Gyrobicupola (Miller, J. C. P. "Polyhedron." Encyclopædia Britannica, 11th ed. [PAGES? YEAR? EDITION?])

Enriques Surface (REQ-102006 ILL REPORTS NOT IN TABLE OF CONTENTS: Enriques, F. "Sulla classificazione." Atti Accad. Naz. Lincei 5, 1914.)

Envelope Theorem (Darboux and Zermelo in 1894; Kneser in 1898)

Equal Parallelians Point (ETC: (ab+ac-bc)(3a^2b^2-2a^2bc-6ab^2c+3a^2c^2-4abc^2+b^2c^2)/a)

Euler Formula (geometric interpretation of complex numbers proof from Conway and Guy;)

Euler's Graeco-Roman Squares Conjecture (Jahrb. Math. 48, 71, 1921.; Jahrb. Math. 49, 41-42, 1923. Bose, Shrikhande, and Parker 1959)

Euler's Homogeneous Function Theorem (Euler, L. Nova Acta Acad. Petropol. 7, p. 58, 1778.)

Euler-Mascheroni Constant (Odena 1982-1983;)

Euler-Mascheroni Constant Approximations (Odena 1982-1983)

Euler's Sum of Powers Conjecture (Letac, A. Gazetta Mathematica 48, 68-69, 1942.)

Exterior Angle Theorem (check Dunham p. 41)

Faà di Bruno's Formula (Dederick, Wall initials;)

Fagnano's Point (figure)

Feigenbaum Constant (sloane: accurate values for Gamma=2.637?;)

Fermat's Last Theorem (Genocchi 1852; Mirimanoff 1905; Lebesgue 1840; Vandiver 1914; Vandiver, H. S. Fermat's Last Theorem and Related Topics in Number Theory. Ann Arbor, MI [PUBLISHER?]: 1935; L'Intermédiaire des Math. 15, 217-218.; Swinnerton-Dwyer, P. Nature 364, 13-14, 1993)

Fermat Number (Fermat 1650; Eisenstein 1844; Keller, W. "Factors of Fermat Numbers and Large Primes of the Form k·2^(n)+1, II." In prep.)

Fermat Prime (Eisenstein 1844)

Fibonacci Number (Hoggatt III+IV)

First Lemoine Circle (Lemoine. Assoc. Français pour l'avancement des Sci. 1873.)

Four Travelers Problem (figure)

Frenet Formulas (figure)

Fuhrmann's Theorem (displace b'; Fuhrmann reference)

Gauss's Circle Problem (Sierpinski 1906; van der Corput 1923)

Gauss's Interpolation Formula (Abramowitz and Stegun don't show if it is plus or times 2 terms)

Gauss's Theorema Egregium (is syntax -R(v,w)v,w correct?)

Geometric Construction (Adler, A. Theorie der geometrischen Konstruktionen.; Enriques, F. Fragen der Elementargeometrie, Vol. 2.; Vahlen, K. T. Konstruktionen und Approximation.)

Geometric Homology (Lambert, J. H. Freie Perspective, 2nd ed. Zürich, Switzerland, 1774.)

Girard's Spherical Excess Formula (Girard, A. Invention nouvelle en algebra. Amsterdam, Netherlands, 1629.)

Giuga Number (Check the date and version of Kellner's 10 July 2003 preprint cited in the bibliography.)

Glaisher-Kinkelin Constant (Glaisher 1877)

Global C(G;T) Theorem (make title bold italic)

Goldbach Conjecture (Desboves, A. Nouv. Ann. Math. 14, 293, 1855. [can't find as cited];)

Green's Function--Poisson's Equation (check derivation in arfken)

Gröbner Basis (Stoutemyer 1984: proceedings editor and publisher not verified.)

Hammer's X-Ray Problems (Croft references)

Hard Hexagon Entropy Constant (Katzenelson and Kurshan (1986): verify the proceedings editors, publication city, and publisher.)

Hardy-Littlewood Tauberian Theorem (Hardy, G. H. and Littlewood, J. E. Quart. J. Math. 46, 215-219, 1915. Hardy, G. H. and Littlewood, J. E. Acta Math. 41, 119-196, 1918.; Karamata. Math. Z. 32, 319-320, 1930.)

Harmonic Mean Index (why is p_0q_0 summed over? Seems strange, but Kenney and Keeping do it)

Harmonic Series (more terms of 2.163)

Hartley Transform (modify discrete case to use plus sign convention)

Hausdorff Paradox (add some references; this entry is too vague; no references in Le Lionnais)

Heart Curve (sloane: 3.661972725)

Helson-Szegö Measure (what is bar y?)

Hempel's Paradox (details; Hempel, C. G. "A Note on the Paradoxes of Confirmation." Mind 55, 1946)

Heptagonal Triangle ([VERIFY INTENDED AMM PROBLEM NUMBER?] "Problem E681" is not by Bankoff and Garfunkel; Amer. Math. Monthly metadata give E681 in 52, 395, 1945, with solution in 53, 157, 1946.)

Hessian Covariant (describe symbols)

Hexyl Triangle (ETC: 3a^3-ba^2-ca^2-3b^2a-3c^2a+2bca+b^3+c^3-bc^2-b^2c)

Hilbert's Axioms (add: actually include all 21 explicitly; http://www.math.uncc.edu/ droyster/math3181/notes/hyprgeom/node27.html http://www.math.uncc.edu/ droyster/courses/spring02/classnotes/Chapter02.pdf)

Hirota Equation (should abs(u squared) be (abs u)squared?)

Hofstadter Ellipse (ETC: center; ETC: fourth intersection with circumcircle)

Hofstadter Triangle (figure)

Hölder's Inequalities (Hölder, O. "Über einen Mittelwertsatz." Göttingen Nachr., 44, 1889.)

Icosahedral Equation (Klein, F. "Sull' equazione dell' Icosaedro nella risoluzione delle equazioni del quinto grado [per funzioni ellittiche]." Reale Istituto Lombardo, Rendiconto, Ser. 2 10, 1877)

Ideal (David Vanvactor recommends adding a discussion of representation and multiplication of ideals)

Idempotent Number (add references listed in Sloane sequences)

In-and-Out Curve (reference)

Incircle (analytically compute Well's theorem)

Inclusion-Exclusion Principle (da Silva. "Proprietades geraes." J. de l'Ecole Polytechnique, cah. 30.)

Information Dimension (Balatoni, J. and Renyi, A. Pub. Math. Inst. Hungarian Acad. Sci. 1, 9, 1956.)

Invariable Point (figure)

Inversion (Thomson, W. "Extrait de deux lettres adressées à M. Liouville." J. math. pures appl. 12, 256, 1847.)

Isoclinic Groups (truncated last sentence?)

Isodynamic Points ([FINAL PAGES? DOI OR PUBLISHER URL? for Rabinowitz and Suppa 2026])

Isogonal Line (Label figure)

Isolated Singularity (check table)

Isomorphic Factorization (Exoo 2005; Royle 2004)

Isosceles Tetrahedron (Chefik-Bey. Nouv. Ann. 19, 403, 1880. Monge, G. Corresp. sur l'École Polytech., pp. 1-6, 1809. Jacobi, C. F. A. In Swinden, J. H. Elemente. p. 457, 1834. Lemoine, E. Z. Math. u. Physik 29, 321, 1884. Biddle, D. Mathesis, p. 91, 1931)

Jacobi's Imaginary Transformation (Poisson, S. Mém. de l'Acad. des Sci. 6, 592, 1827.)

Jonquière's Theorem (Jonquière, A. "Mémoire sue les contacts multiples d'ordre quelconque de courbes, etc. [REST OF TITLE?]" J. reine angew. Math. 66, 288 [END PAGE?], 1866;)

Jordan Decomposition Theorem (what does A have to do with anything)

Joyce Sequence (Laisant, C. A. 1906; Uhler 1947)

Julia Set (Douady;)

Keith Number (sloane: 71 and 1593583.)

Kelvin's Conjecture (Matzke, E. Amer. J. Botany 32, 130, 1946.; Princen, H. M. and Levinson, P. J. Colloid Interface Sci. 120, 172, 1987.; Ross, S. Amer. J. Phys. 46, 513, 1978.; Thomson, W. Philos. Mag. 25, 503, 1887.; Weaire, D. Philos. Mag. Let. 69, 99, 1994.; Williams, R. Science 161, 276, 1968.)

Kieroid (plot)

Kirkman Points (Hesse (1861), p. 186 [IS THIS THE RIGHT REFERENCE?])

Kirkman's Schoolgirl Problem (RARE BOOK: Kirkman, T. P. Lady's and Gentleman's Diary. 1850. Woolhouse. Lady's and Gentleman's Diary. 1862-1863. Carpmael. Proc. London Math. Soc. 12, 148-156, 1881.)

Kleinian Group (add figure)

Knight Graph (Jaenisch. Chess Monthly 1859. Roget, P. M. Philos. Mag. 16, 305-309, 1840. Warnsdorff, H. C. Des Roesselsprungs einfachste und allgemeinste Loesung. Schmalkalden, Germany, 1823.)

Knot (ftp://chs.cusd.claremont.edu/pub/knot; Schubert, H. Sitzungsber. Heidelberger Akad. Wiss., Math.-Naturwiss. Klasse, 3rd Abhandlung. 1949; Przytycki, J. "A History of Knot Theory from Vandermonde to Jones." Proc. Mexican Nat. Congress Math., Nov. 1991.)

Kobon Triangle (Do these appear in Kobon Fujimura's The Tokyo Puzzles. Frederick Muller Limited, 1981 New York, Charles Scribner's 1978 (?) (Gardner does not say.))

Kolakoski Sequence (Sellke [FIRST INITIAL?]. Problem 324 in Statistica Neerlandica 50, 222-223, 1996.)

Kollros' Theorem (Kollros 1938)

Lagrange Bracket (Lagrange. Mém. de l'Institute de France, 1808. Reprinted in Oeuvres, Vol. 4. p. 713)

Landau Symbols (Bachmann, P. Analytische Zahlentheorie. 1894. Landau, E. 1909)

Langberg-Médard Conjecture (Langberg and Medard 2009 Allerton proceedings: editors and publication city?)

Lattice Sum (add definition)

Leading Order Analysis (define psi function)

Leg (figure)

Legendre's Constant (Abel, N. H. Letter to Holmboe. 4 August 1823. In Niels Henrik Abel, Memorial, Publié à l'Occasion du Centenaire de sa Naissance. [PAGES? YEAR? PUBLISHER?])

Lemoine Cubic (Lemoine, E. [TITLE?] Association Française pour l'Avancement des Sciences. [VOLUME?] p. 149 [PAGE RANGE?], 1891.; Lemoine, E. [TITLE?] Association Française pour l'Avancement des Sciences. [TITLE?] p. 124 [PAGE RANGE?], 1892.;)

Leonard Graph ([TITLE?] Leonard (1979) thesis.)

Liar's Paradox (Fraenkel, A. A. and Bar-Hillel, Y. Foundations of Set Theory. Amsterdam, Netherlands, p. 11, 1958.; Weyl, H. Philosophy of Mathematics and Natural Science. Princeton, NJ, p. 228, 1949)

Local Density (reference. Stewart?)

Logarithmic Integral (Soldner. [TITLE?] Abhandlungen 2, 333 [ENDING PAGE?], 1812.)

Logarithmic Spiral Catacaustic (find exact relation between original and catacaustic)

Logistic Map (Andrecut 1998; Gonzáles and Pino 1999, 2000; Gonzáles et al. 2001ab; Umeno 1998; Wong et al. 2001;)

Lommel Function (Lommel, E. C. J. von. "Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisrunden Schirmchens theoretisch und experimentell bearbeitet." Abh. der math. phys. Classe der k. b. Akad. der Wiss. (München) 15, 229-328, 1886.: page range listed as 229-328 and 233-328 on pp. 537 and 772 of Watson, resp.; Lommel, E. C. J. von. "Die Beugungserscheinungen geradlinig begrenzter Schirme." Abh. der math. phys. Classe der k. b. Akad. der Wiss. (München) 15, 531-663, 1884-1886.: page range listed as 529-664 and 531-663 on pp. 537 and 772 of Watson, resp.;)

Longimeter ()

Longuet-Higgins Circle (Woo, P. Y. "Solution of Problem 10734." Amer. Math. Monthly [VOLUME? PAGES? YEAR?])

Longuet-Higgins Point (Woo, P. Y. "Solution of Problem 10734." Amer. Math. Monthly [VOLUME? PAGES? YEAR?])

Löwenheim-Skolem Theorem (Skolem, T. "Sur la portée du théorème de Löwenheim-Skolem." Les Entretiens de Zurich sur les fondements et la méthode des sciences mathématiques (December 6-9, 1938), pp. 25-52, 1941. Quine, W. V. Methods of Logic, rev. ed. New York: pp. 253-260, 1959.)

Macbeath Circle (ETC: X933^'=((b-c)^2*(b+c)^2*(a^8-2*a^6*b^2+2*a^2*b^6-b^8-2*a^6*c^2+a^4*b^2*c^2+b^6*c^2+2*a^2*c^6+b^2*c^6-c^8)*Cos[A]*Cos[B-C])/a)

Macbeath Inconic (Serret, P. [TITLE?] Nouv. Ann. Math. [VOLUME?], p. 428 [PAGE RANGE?], 1865.)

Maclaurin Series (sloane: add OEIS numbers)

Magic Tesseract (reconcile existence of order 5 magic tesseract)

Magic Tour (Beverly, W. Philos. Mag. p. 102, Apr. 1848.; Murray, H. J. R. The Magic Knight's Tours, a Mathematical Recreation. [PUBLISHER, CITY?] 1951.; Wenzelides, C. Schachzeitung, p. 247, 1849.)

Malfatti's Problem (Hart, [INITIAL?]. [TITLE?] Quart. J. 1, p. 219. [YEAR?]; Oswald, [INITIAL?]. [TITLE?] Klassiker de exakten Wissenschaften, Vol. 23. Suppl. [PAGES?] [YEAR?]; Schellbach, [INITIAL?]. [TITLE?] J. reine angew. Math. 45 [PAGES?] [YEAR?].)

Mandart Inellipse (find area and axes)

Mandelbrot Set (Mandelbrot; Douady, A. "Julia Sets and the Mandelbrot Set." In The Beauty of Fractals: Images of Complex Dynamical Systems (Ed. H.-O. Peitgen and D. H. Richter). Berlin, Germany: Springer-Verlag, pp. 161-?.; Fisher, Y. and Hill, J. "Bounding the Area of the Mandelbrot Set." Submitted to Numer. Math.; Hill, J. R. "Area of Mandelbrot Set Components and Clusters.")

Mannheim's Theorem (figure; Mannheim, A. J. de math. élémentaires, p. 225, 1895.)

Markov Chain (Discuss transition probabilities)

Mascheroni Construction (Mascheroni, L. Geometry of Compass. 1797)

Meixner Polynomial of the Second Kind (discrepancy between Roman and me for G.F.)

Mellin-Barnes Integral (Pincherle, S. Atti d. R. Academia dei Lincei, Ser. 4, Rendiconti 4, 694-700 and 792-799, 1888)

Menelaus' Theorem (check)

Mersenne Prime (According to Schroeppel, a British guy "Hayworth, Guy" has a nice memo about discovery and proofs of Mersenne numbers; Miller, J. C. P. and Wheeler, D. J. Nature 168, 838, 1951; Landry. 1869. Lehmer. 1947. Penk and Baillie. 1979. Poulet. 1923 Sandia 1984 Pomerance and Wagstaff Sci News Euler. 1750. Lehmer. 1952-3. Pervouchine. 1883. Seelhoff. 1886. Slowinski, D. Jan. 29, 1988. Slowinski, D. Sci. News 139, 191, 9/16/1989)

Mertens Function (add sloane sequences from table)

Mice Problem (Clapham, A. J. [TITLE?] Rec. Math. Mag., [VOLUME, PAGES?] Aug. 1962.;)

Minkowski Convex Body Theorem (Warmus, W. Colloq. Math. I 1, 45-46, 1947.)

Miquel's Theorem (Kimberling, C. "Transfigured Triangle Geometry." Preprint. Mar. 5, 2005.)

Möbius Function (Landau, E. Wiener Sitzungsber. 120, 973-988, 1911.; Landau, E. Prac. Matematyczno-Fizycznych 21, 97-177, 1910.)

Möbius Strip (Listing and Tait. Vorstudien zur Topologie, Göttinger Studien, Pt. 10, 1847.; Möbius, A. F. Werke, Vol. 2. p. 519, 1858.)

Mock Theta Function (Bellman (1961, p. 51), the first term in the Denominator is (erroneously?) written (1+q)^2)

Moduli Space (Kirwan, F. "Introduction to Moduli Spaces." In Proceedings of the EWM Workshop on Moduli Spaces, Oxford, EWM. 1999.)

Monge Point (figure; Monge, G. Corresp. sur l'École Polytech. 2, 266, 1795.)

Monge's Tetrahedron Theorem (figure; Forder, H. G. Math. Gaz. 15, p. 470, 1930-1931. Monge, G. Corresp. sur l'École Polytech. 2, 266, 1795.)

Moore Graph (sloane: 0,0,0,1,1,2,1,2,1,3,1,2,1, ...[too many sequences match the initial nonzero terms])

Morse Theory (Rassias, G. (Ed.). Morse Theory and Its Applications.)

Moses-Longuet-Higgins Circle (ETC: (a^2(S_(A)-4s_(a)^2)+4(s_(c)^2S_(B)+s_(b)^2S_(C)))/a)

Mott Polynomial (inconsistency in Roman)

Multicommodity Flow (Langberg and Medard 2009 Allerton proceedings: editors and publication city?)

Multiperfect Number (Carmichael and Mason, T. E. Proc. Indian Acad. Sci., 257-270, 1911)

Napoleon's Problem (Mascheroni, L. Geometria del compasso. 1797. Schwerin. Mascheronische Konstruktionen. 1898.)

Nested Radical (Ramanujan, S. Question No. 298. J. Indian Math. Soc. [VOLUME, PAGES?] 1911.)

Noncentral F-Distribution (Bulgren, W. G. "On Representations of the Doubly Non-Central F Distribution." J. Amer. Stat. Assoc. 66, 184, 1971)

Number (Frege. Grundgesetze der Arithmetik.)

Ochoa Curve (ILL: not found as cited: Crux 1987 248; ILL not found as cited: Crux 1989 41; Ochoa Melida, J. "La ecuacion diofántica b_0y^3-b_1y^2+b_2y-b_3=z^2." Gaceta Math. 139-141, 1978.)

Odd Sequence (missing definition)

Operations Research (INFORMS. Operations Research: 50th Anniversary Issue. Linthicum, MD, 2002.)

Orchard-Planting Problem (Educ. Times Reprints 8, 106, 1868. Educ. Times Reprints 45, 127-128, 1886.)

Orthocentric System (add labels)

Orthopolar Line (Servais, Cl. "Sur l'orthopóle." Mathesis, p. 11, 1923.)

Oval (Dixon, R. "The Drawing Out of an Egg." New Sci., July 29, 1982.)

Pandigital Fraction (Friedman reference: [TITLE? PAGES? YEAR?])

Pascal's Theorem (Cayley, A. Quart J. 9, p. 348.; Leibniz, G. Letter to M. Périer. In Œuvres de B. Pascal, Vol. 5 (Ed. Bossut). p. 459. Veronese. "Nuovi Teremi sull' Hexagrammum Mysticum." Real. Accad. dei Lincei. 1877.)

Pathological (Poincaré 1899)

Pearson System (locate type 8)

Pedal Line (figure)

Pell Equation (Nasimoff, P. S. Ch. 1 in Application of Elliptic Functions to the Theory of Numbers. Moscow, Russia, 1885. French summary in Ann. sci. de l'École normale supér. 5, 23-31, 1888.; Euler, L. Novi Comm. Acad. Petrop. 18, 218, 1773. Reprinted in Opera Omnia, Vol. 3, p. 310. Euler, L. Comm. Arith. 570. Reprinted in Opera Omnia, Vol. 3, p. 310.)

Perfect Number (Sylvester (1887). Catalan (1888). Stuyvaert (1896). (Lucas 1891) Reference and date for Ruiz floor-sum criterion need verification.)

Perfect Square Dissection (Reichert, H. and Toepken, H. Jahresber. deutschen math. Verein. 50, 1940.; Willcocks, T. H. Fairy Chess Review 7, 1948.;)

Periodogram (Schuster. Terrestrial Magnetism 3, 24, 1898)

Perspective (Lambert, J. H. Freie Perspective, 2nd ed. Zürich, Switzerland, 1774.)

Pi Approximations (Castellanos 1988a or b)

Pi Continued Fraction (Posting date of the Stoschek source remains unverified; the earliest verified archive capture is Dec. 3, 2000. Stern (1833, p. 268, equation 6) prints inconsistent plus signs after its initial two minus signs. The all-minus expansion follows his formula 4 on p. 267; a historical erratum has not been located.)

Pi Heptomino (new figure)

Pi Iterations (Pfaff 1800)

Pick's Theorem (Steinhaus, H. "O polu figur płlaskich." Przeglad Mat.-Fiz., 1924.; Haigh, G. "A 'Natural' Approach to Pick's Theorem." Math. Gaz. 64, 173-, 1980.)

Plaited Polyhedron (Gorham, J. Plaited Crystal Models. 1888.)

Plancherel's Theorem (Plancherel 1910;)

Plane Curve (Archibald, R. C. "Curves." Encyclopædia Britannica, 14th ed. [VOLUME? PAGES? YEAR?])

Plane Division by Circles ([AUTHORS of Problem Q736, Parabola 1988, and the Indiana School Mathematics J. 1979 reference?])

Plane Geometry (Fuhrmann, W. Synthetische Beweise Planimetrische Sätze. Berlin, Germany, 1890.)

Plouffe's Constants (Chowdhury, M. "On Iterates of the Chaotic Logistic Function t_(n+1)=4t_(n)(1-t_(n))." Unpublished note, 2001.; Chowdhury, M. "A Formula for 0.4756260767...." Unpublished note, 2001.)

Plücker Lines (Plücker, M. J. reine angew. Math. 5, p. 274.)

Point-Set Topology (Bing, R. H. "Elementary Point Set Topology." Amer. Math. Monthly 67, No. 7, Aug.-Sep. 1960.)

Poisson Trials (Uspensky)

Pollard rho Factorization Method (Eldershaw, C. and Brent, R. P. "Factorization of Large Integers on Some Vector and Parallel Computers." [JOURNAL, VOLUME, PAGES, YEAR?])

Pólya-Vinogradov Inequality (Vinogradov. [TITLE?] Perm. Univ. Fiz.-Mat. ob.-vo Zh. 1, 18-24 and 94-98, 1918.)

Polygon Diagonal Intersection Graph (Meeus, J. [TITLE?] Wiskunde Post (Belgium) 10, 62-63, 1972.)

Polyhe (add figures)

Polyhedral Formula (Euler, L. "Elementa doctrine solidorum." Reprinted in Opera, Vol. 26, pp. 71-92. [PUBLISHER, CITY, YEAR])

Polyomino (Applied Combinatorial Mathematics, pp. 201 and 213;)

Poncelet's Porism (Darboux, G. C. R. Acad. Sci. Paris 90, 1880.; Darboux, G. Principles de géométrie analytique, Vol. 3. Paris, France, pp. 250-287, 1917.; Davis, M. A. Educ. Times 32.; Durége. Theorie der Elliptischen Functionen. p. 185.; Educ. Times 3, p. 42.; Fuss, N. Nova Acta Petropol. 10, 1792.; Hart. Quart. J. Math., 1857.;)

Pong Hau K'i (What does the board actually look like and what in meant by finite graph)

Prime Constellation (sloane: 2.640647264; sloane: 2.858248596; sloane: 4.151180864; sloane: 8.302361728)

Prime-Generating Polynomial (sloane: -18757, -15619, -12829, -10369, -8221, -6367, -4789, -3469, -2389, -1531, -877; sloane: 50383, 45779, 41381, 37189, 33203,; sloane: 2971, 2477, 2069, 1747, 1511, 1361,; etc.)

Prime Number (references from Hardy and Wright p. 414; Gandhi, J. M. "Formulae for the Nth Prime." Proc. Washington State University Conferences on Number Theory. pp. 96-107, 1971. Guy, R. K. "Divisors and Desires." Amer. Math. Monthly 104, 359-360, 1997.)

Prime Number Theorem (Bochner. Math. Z. 37, 1-9, 1933. Landau, E. Berliner Sitzungsber., 514-521, 1932.; Lu, W. C. "On the Elementary Proof of the Prime Number Theorem with a Remainder Term." Rocky Mountain J. Math. 29, 979, 1999. Vallée Poussin, C. Mém. Couronnés Acad. Roy. Belgique 59, 1-74, 1899.;)

Prime Quadratic Effect (Hardy and Littlewood 1915(?))

Prime Triangle (Kenney, M. J. "Student Math Notes." NCTM News Bulletin. Nov. 1986)

Projective Geometry (Poncelet, J.-V. Traité des Propriétés Projectives. Paris, France, 1822. Reye. Geometrie der Lage, 2nd ed. Hannover, Germany, 1877. Steiner, J. Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander. Berlin, Germany, 1832.)

Prüfer Ring (Fontana, M.; Huckaba, J. A.; and Papick, I. J. Prüfer Domains. New York: Dekker.)

Puzzle (references; Clessa, J. J. Math and Logic Puzzles for PC Enthusiasts. New York: Dover. Costello, M. J. The Greatest Puzzles of All Time. New York: Dover.)

Pythagorean Triple (check Honsberger: Garfunkel, J. Pi Mu Epsilon J. 1981, p. 31.)

q-Dimension (Grassberger, P. "Generalized Dimensions of Strange Attractors." Phys. Lett. A 97, 227, 1983. Hentschel, H. G. E. and Procaccia, I. "The Infinite Number of Generalized Dimensions of Fractals and Strange Attractors." Physica D 8, 435, 1983.)

Quadrant (label axes)

Quadratic Field (Berg (1935): title and original bibliographic metadata? Oppenheim (1934): author initials and title?)

Quasigroup (numbers for each order)

Radial Curve (figure)

Ramanujan's Square Equation (Nagell (1948): title of the paper in Norsk Mat. Tidsskr. 30, 62-64?)

Random Number (Janson Random number generators)

Rational Double Point (19 out of 20)

Reciprocal Curve (Relation to the inverse curve?)

Regular Heptagon (Thébault, V. Mathesis, p. 204, 1913.; Thébault, V. Mathesis, p. 169, 1938.; Thébault, V. Mathesis, p. 344, 1950.; Thébault, V. Mathesis, p. 78 and 329, 1955.; Thébault, V. Mathesis, p. 106 and 149, 1956.)

Relatively Prime (Cesaro and Sylvester 1883; Sylvester 1909)

Reversal (Kraitchik, M. La Mathématique des Jeux. Brussels, Belgium, pp. 55, 59, 1930.;)

Riemann-Siegel Functions (Odlyzko, A. M. "The 10^(20)th Zero of the Riemann Zeta Function and 70 Million of Its Neighbors." Preprint.;)

Riemann Zeta Function (Check Stark 1974 formula (37); Odlyzko, A. M. "The 10^(20)th Zero of the Riemann Zeta Function and 70 Million of Its Neighbors." Preprint.;)

Riemannian Geometry (Previato: Much of the treatment of Riemannian geometry relies on books address to engineers)

Rotor (Uspekhi Mat. Nauk 3, No. 2 (24), 239, 1948)

Roundoff Error ([AUTHOR, TITLE?] The Toronto Star. [PAGES?] November 19, 1983.)

Rule 62 (sloane: 1, 7, 25, 111, 433, 1751, 7033, 28047, 112305; sloane: 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1)

Rule 126 (sloane: 1, 7, 27, 127, 387, 1935, 6579, 32767; [not via superseeker] sloane: 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0,)

Rule 182 (sloane: 1, 7, 21, 127, 381, 1911, 5461; [not via superseeker] sloane: 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1,)

Schlömilch Remainder (Roche. Mem. de l'Acad. de Montpellier. 1858.)

Schlömilch's Series (Chapman. Quart. J. 43, 34-37, 1912.; Schlömilch. Z. für Math. Phys. 3, 137-165, 1857.)

Schoute's Theorem (Not found as cited (two times): Schoute, P. H. Proc. Amsterdam Acad., 39-62, 1887-1888)

Seiffert's Spherical Spiral (Seiffert. "Über eine neue geometrische Einführung in die Theorie der elliptischen Funktionen." Wissensch. Beiträge Jahresber. Städtischen Realschule zu Charlottenburg, Ostern. 1896.)

Self Number ([TITLE?] Joshi (1973) dissertation.)

Semiperimeter (add dotted lines)

Separation Theorem (the function should be specified as well as what the lambda_(v) represent.)

Sierpiński's Composite Number Theorem (Keller, W. "Factors of Fermat Numbers and Large Primes of the Form k·2^(n)+1, II." Preprint available at http://www.rrz.uni-hamburg.de/RRZ/W.Keller/)

Similar (Neuberg and Tarry. Mathesis 2.)

Simple Function (add diagram graph of a simple function)

Sine-Triple-Angle Circle (Tucker and Neuberg, J. Mathesis 12, 1887. [Cited as "M, 1887-12"])

Siteswap (looks underlined after latex2html gets through with it!)

Skewes Number (Asimov, I. "Science: Skewered!" Mag. Fantasy Sci. Fiction [VOLUME, PAGES?] Nov. 1974.)

Smith's Markov Process Theorem (find a reference and check)

Soddy Circles (Identify the intended Martini (1994) chapter and verify its title and page range in Geometrie und ihre Anwendungen. Check whether the purchasing link refers to the cited Giering-Hoschek collection.)

Solid of Revolution (ADD: Barrels by cylinders)

Spencer's Formula (Spencer, J. J. I. A. 38, 334, 1904. Spencer, J. J. I. A. 38, 339, 1904. Spencer, J. J. I. A. 41, 361, 1907.)

Sperner's Lemma (add figure)

Spherical Covering (Hardin, R. H.; Sloane, N. J. A. S.; and Smith, W. D. Spherical Codes. In preparation.)

Standard Map (Hermann; Italians; Greene; Mather)

State Diagram (Abbott, R. "The Farmer Goes to Market." Sci. Amer. [Volume?], [Page range?], Oct. 1962.)

Stieltjes Integral (Pollard, S. Quart. J. Math. 49, 73-138, 1923.;)

Stirling's Approximation (Stirling, J. Methodus differentialis. 1730.)

Strophoid (add figure)

Student's t-Distribution (Shaw, W. T. "Sampling Student's T Distribution--Use of the Inverse Cumulative Distribution Function." J. Comput. Finance 9, 37-73, 2006.)

Sturm Function (Sturm, C. "Mémoire sur la résolution des équations numériques." Bull. des sciences de Férussac vol. 11, 1929. quoted in Dörrie p. 112)

Subresultant (Hong 1999: later publication status and current URL unresolved.)

Sum of Squares Function ([Verify the Shanks (1993) page range: the former range 162-153 was descending.])

Tau Function (Darling, H. B. C. Proc. London Math. Soc. 19, 350-372, 1921. Lehmer 1949 Ramanujan, S. Proc. London Math. Soc. 18, 1920. Swinnerton-Dyer 1973)

Taylor Circle (Taylor, H. M. Proc. London Math. Soc. 15.)

Taylor's Theorem (Taylor 1715 reference; Gregory 1670-1680 reference)

Tetrahedron (Wells, D. "Puzzle Page." Games and Puzzles. Sep. 1975.)

Tetrix (change rotation and scaling of third figure to be same as first two)

Thābit ibn Qurra Rule (Gérardin Sphinx-Oedipe 1907-8, 49-56, 65-71; Gérardin Assoc. Franç avanc. sc. 37, 1908, 36-38; Gérardin l'intermédiaire des math. 1909, 104 Woepcke, F. J. Asiatique 20, 320-429, 1852.;)

Thomson Cubic (Thomson, F. D. [TITLE?] Educ. Times. [PAGES?] [ISSUE/VOLUME?] Aug. 1864.)

Tiling (Gardner, M. Penrose Tiles to Trapdoor Ciphers. Washington, DC: Math. Assoc. Amer.; Abas)

Torsional Rigidity (Dinnik, A. [TITLE?] Bull. Don. Polytech. Inst. Vovotcherkassk 1, 309 [ENDING PAGE], [YEAR?].)

Triangle Geometry (Kimberling, C. "Transfigured Triangle Geometry." Preprint. Mar. 5, 2005.)

Triangular Number (Cesaro 1886)

Trilinear Coordinates (Wong, M. K. F. Int. J. Math. Educ. Sci. Tech. 27, 293-296, 1996. Wong, M. K. F. Int. J. Math. Educ. Sci. Tech. 29, 143-145, 1998.)

Truchet Tiling (are the exact circle and dumbbell fractions known?)

Tschirnhausen Transformation (Tschirnhaus. Acta Eruditorum. 1683.)

Twin Prime Cluster (Mudge, M. "Some Sequences of Smarandache Contrasted with a Permutations Problem Relating to Class/student Allocation." Personal Computer World, London, England. [VOLUME?] pp. 674-675, June 1995.; Mudge, M. "Smarandache Sequences and Related Open Problems." Personal Computer World. [VOLUME, PAGES?] Numbers Count, February 1997.)

Twin Primes (Ricci; Pil'Tai 1972; Atkin and Richert. Amer. Math. Soc. Not. 26, A-373, 1979. Ruiz. To be published in Math. Gaz. 2000 Glaisher 1878)

Uniform Convexity (add figure a la Cheney 1966, p. 21, with two end points on an ellipse (which is uniformly convex) and on an octagon, (which isn't))

Vertex Triangle (Kimberling, C. "Transfigured Triangle Geometry." Preprint. Mar. 5, 2005.)

von Mises Distribution (any idea how to calculate the mean or circular variance?)

Voting Paradoxes (Lipton, R. G.; and Widgerson, A. "Multi-Party Cryptographic Protocols."; Riker, W. H. "Voting and the Summation of Preferences." Amer. Political Sci. Rev., Dec. 1961. Saari, D. G. Math. Intell. 10, 32, 1988.)

Wavelet (REWRITE)

Weill Point (Weill. Liouville's J. (Ser. 3) 4, 270, 1878.)

Weill's Theorem (Casey, J. Quart. J. Pure Appl. Math. 5, 44, 1862.; Weill. Liouville's J. (Ser. 3) 4, 270, 1878.)

Weyrich's Formula (check this; it doesn't seem right)

Wilkie's Theorem (check formula)

Wilson Polynomial (Check Koepf's definition)

Wilson Prime (Gonter, R. H. and Kundert, E. G. "All Numbers Up to 18876041 Have Been Tested without Finding a New Wilson Prime." Preprint, 1994.)

Zebra Graph (Jelliss, G. Chessics. [VOLUME? PAGES? YEAR?])

Zonohedron (Fedorov, E. S. Zeitschr. Krystallographie und Mineralogie 21, 689, 1893.)