 Pell's equation
 MacTutor Math History Archives
Linked essay on the indeterminate quadratic equation. With references and other related web sites.
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 Pennies in a Tray  Ivars Peterson (MathLand)
What's the largest number of pennies that you can pack inside a circular tray to form a carpet of nonoverlapping coins? What about inside a square or triangular tray? What if you could expand or contract the size of all your pennies to fit a required
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 A Penny Surprise  Ivars Peterson (MathTrek)
Flipping a coin in the air, catching it, then determining whether it has come up heads or tails is to many people a prototypical random process. In some sense, however, coin tossing isn't really random at all. A mechanical gadget can flip a properly positioned
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 Pens and Crystals (Math Chat)  Frank Morgan, MAA Online
Answering the challenge: How should you fence in two adjacent rectangular pens of given areas A, B with the least amount of fencing (including the fencing separating them)? Ms. Mattuck's 8th graders at the Richmond School use the Geometer's Sketchpad
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 Pentium Bug Revisited  Ivars Peterson (MathLand)
The incident that triggered a barrage of ridicule occurred in the fall of 1994, a few months after Intel had introduced its Pentium microprocessor. The furor started with an email message from Thomas R. Nicely, a mathematician at Lynchburg College in
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 A Perfect Collaboration  Ivars Peterson (MathLand)
Euclid and Euler seem an unlikely pairing. One was the most prominent mathematician of antiquity, best known for his treatise on geometry, the Elements. The other was the most prolific mathematician in history, the man whom his 18century contemporaries
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 Perfect Numbers  MacTutor Math History Archives
Linked essay describing the work on perfect numbers from Pythagoras through Euclid, Nicomachus, the Arabs, Cataldi, Descartes, Mersenne, Fermat, Euler, Barlow, Catalan, Cole, and others; with other web sites and 20 references (books/articles).
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 Performance Evaluation  Keith Devlin (Devlin's Angle)
"Dear Professor G,..." A critique of current professorial evaluation schemes in the college/university setting.
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 Periodic Points of Quadratic Polynomials  Cut the Knot!, Alexander Bogomolny
The main purpose of the Mandelbrot set is to index Julia sets corresponding to various values of the parameter c... iterations inside M evolve differently depending on the value of c. (You may explore some iterations via the applet supplied.) An interactive
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 The perplexing mathematics of presidential elections  Keith Devlin (Devlin's Angle)
It's not the idea of one person one vote that's the problem, it's that math that is used to turn those votes into a final decision. Ideally, that math should reflect the wishes of the electorate. But does it? The answer usually comes as a surprise to
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 PetaCrunchers  Ivars Peterson  Science News Online
Setting a course toward ultrafast supercomputing: floating point calculations have become so integral to computers that their performance is often measured in floating point operations per second (FLOPS). The next logical step is petaFLOPS computing,
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 Pete Casazza
Articles by this University of Missouri professor, some of them scanned from journals, include "A Mathematician's Survival Guide," "Classes of Finite Equal Norm Parseval Frames," "Dear Pythagoras: See What You Started," "Equivalents of the KadisonSinger
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 Pete L. Clark
Course notes, open problems, publications and preprints by Clark, who teaches at the University of Georgia, and does research primarily in number theory and arithmetic geometry. See, in particular, Clark's freely downloadable PDF expositions on commutative
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 Peter Doyle  Mathematics, Univ. of California at San Diego
Advanced tutorials, proofs and papers, mostly statistics, probability, geometry, and graph theory, but with examples from other mathematics areas as well. Don't miss Dr. Doyle's Flip Flops.
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 Peter J. Lu  Department of Physics, Harvard University
Lu's mathematical and historical research has revealed new insights in medieval Islamic quasicrystalline tilings and the earliest machines, such as the one he suggests precisely interconverted the linear and rotational motion responsible for the distinctive
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 peterlu.org  Peter Lu
Publications and research into medieval Islamic geometry (e.g., decagonal and quasicrystalline tilings in its architecture), softmatter and colloid physics, highperformance imaging, ancient Chinese technology (e.g., first use of diamond, early precision
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 Peter Olver's Home Page  Peter Olver
Publications, seminar and conference talks, and more resources from the Head of the University of Minnesota's School of Mathematics. Download notes of Olver's lectures on applied mathematics, numerical analysis, orthogonal bases and the QR algorithm,
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 Petits Textes sur l'éducation  Laurent Lafforgue
Essais écrit par le laureat de la medaille Fields 2002, incluent "De l'école et de ce qui fonde la valeur de la culture et du savoir" et "De la culture générale des futurs instituteurs et professeurs" et "Les mathématiques
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 Pfaff's Method (III): Comparison with the WZ Method (Organic Mathematics Proceedings)  George Andrews
In the 1990's, the WZ method has been the method of choice in resolving new conjectures for hypergeometric identities. The object here is to compare the WZ method with Pfaff's method. Such a comparison should (it is hoped) provide some suggestions for
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 Phalanstère  Équipe de Combinatoire Algébrique, Université de MarnelaVallée
Research into the relations between algebra and combinatorics. Membres; Publications; Logiciels; Séminaire. In French.
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 Phase State Diagrams  Harold Brochmann
Phase diagrams reveal much information about physical phenomena. Brochmann illustrates with a variety of graphs created by computer programs. Phase state diagrams of unstable orbits reveal 'strange attractors', useful in a variety of fields including
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 Philippe Flajolet
Philippe Flajolet researches the analysis of algorithms, analytic combinatorics, computer algebra, asymptotic analysis, special functions, random structures, and natural languages. An extensive collection of his articles, and the first chapters of
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 Phi: That Golden Number  Mark Freitag, Univ. of Georgia
Phi can be found in many geometric shapes as an approximation of the Golden Ratio. A geometric figure commonly associated with Phi is the Golden Rectangle, which has sides A and B that are in proportion to the Golden Ratio. It has been said that the Golden
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 PhotoMath  Martha Jean Barrett; Kodak
A project where students created their own mathematical board games for use with younger or less advanced students, for grades 48. Working in teams, advanced students covered seven math categories: whole numbers, decimals, fractions, measurement, geometry,
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 Phyllotaxis  Pau Atela, Chris Golé; Smith College
An interactive site for the mathematical study of plant pattern formation. Sections of the site include: Introduction to Phyllotaxis (classification, Fibonacci/golden angle, microscope, mathematical model, and lattices); history; glossary; references;
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 Physics (MathPages)  Kevin Brown
More than 60 "informal notes" by Kevin Brown on mathematical physics: Doppler for sound and light, optical rays in heterogeneous media, radial orbitgs in Schwarzschild geometry, the formation and growth of black holes, can Schrodinger's cat factor numbers?,
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 The Physics of Baseball  Alan Nathan
Read about the benefits (or not) of corked bats, the differences between aluminum and wood bats, the issues involved with characterizing the performance of bats from a physics point of view, and a description of the collision between a wooden bat and
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 Pi: A 2000Year Search Changes Direction  Adamchik, Wagon
A New Formula for Pi: David Bailey of NASA/Ames and Peter Borwein and Simon Plouffe of the Centre for Experimental and Computational Mathematics at Simon Fraser University (BBP) have discovered a remarkable and remarkably simple new formula for Pi. Sum
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 PianoMath Home Page: Improving Math Skills through Music  Peter Taussig, E. Paul Goldenberg
Essays and activities on links between music and mathematics. PianoMath is
a crossdisciplinary curriculum of musical classroom activities relating to
arithmetic, geometry, graphing, logic foundation, and prealgebra. It is
now being tested at an elementary
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 Pick a Digit, Any Digit  Ivars Peterson (MathTrek)
One of the most amazing mathematical results of the last few years was the discovery of a surprisingly simple formula for computing digits of the number pi. Unlike previously known methods, this one allows you to calculate isolated digits  without computing
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 Picking Winners  Ivars Peterson (MathTrek)
One of the attractive features of spectator sports is the uncertainty of the outcome. Even when one team is overwhelmingly favored to win, the underdog may still come through with a surprising victory. Nonetheless, the ability to pick winners can be of
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 Pick's Theorem  Cut the Knot!, Alexander Bogomolny
A description with proof of Pick's theorem, including links to further pages including exercises with an applet for the geoboard, Farey Series, and SternBrocot Tree. An interactive column for MAA Online that uses a Java applet to simulate a puzzle or
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 Pinpointing Prey  Ivars Peterson (MathTrek)
The sand scorpion uses two types of sensors on the tips of its legs to detect mechanical vibrations transmitted over relatively short distances across the surface. One sensor, called the basitarsal slit sensillum, detects surface waves, or ripples. The
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 PiSearch
Search the first four billion binary digits of Pi for a string. Enter the string in hex (4 bit) or character (5 bit) format. Includes some statistics (probability of occurrence of characters, digits). Related links include articles and papers.
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 Pitfalls of Data Analysis  Clay Helberg
Subtitled "... or How to Avoid Lies and Damned Lies," this paper discusses what people "often overlook in their data analysis, and ways people sometimes 'bend the rules' of statistics to support their viewpoint." Tips to ensure the clarity and accuracy
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 Pi Through the Ages  MacTutor Math History Archives
A history of pi: the Rhind papyrus (Egypt), Ptolemy, Tsu Ch'ung Chi, al'Khwarizmi, Al'Kashi, Viet, Romanus, Van Ceulen, Gregory, Shanks, Lambert, Euler, Buffon, and many more, with other Web sites and 30 references (books/articles).
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 pLab: Random Number Generation  Peter Hellekalek; Mathematics Dept., University of Salzburg
Results and links on this fundamental topic in stochastic simulation. RNGs are deterministic algorithms that produce numbers with certain distribution properties. Roughly speaking, these numbers should behave similar to realizations of independent, identically
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 Place Value  Cut the Knot!, Alexander Bogomolny
Dictionary definitions of place value and what they lack; the emphasis on place value in the NCTM Standards; its importance in understanding our numeration system; how it arises from one of several possible representations of grouping in the counting
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 Plane Isometries  Interactive Mathematics Miscellany and Puzzles, Alexander Bogomolny
A collection of basic facts and partial explanations on plane isometries. With accompanying Java applets for translation, reflection, rotation and glide reflection.
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 Plane Patterns  Ivars Peterson (MathTrek)
Exploring the link between mathematical symmetry and artistic effort. One and twodimensional repeating patterns appear ubiquitously on surfaces, from quilts and colored fabrics to pottery and ceramic tiles... Analyzing the symmetries that underlie such
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 Planes of Budapest  Ivars Peterson (MathTrek)
Nearly every Sunday during the winter of 1933 in Budapest, a small group of students would meet somewhere in the city at a park or cafe to discuss mathematics. The gathering typically included Paul Erdös (19131996), György (George) Szekeres,
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 Planet Math
Designed to be a central repository for mathematical knowledge on the web, with a pedagogical slant. All content is usercontributed, either to the Encyclopedia or to the reviewed listings of papers, books, and lecture notes. Polls, discussions, and feedback
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 Plato: Acceptance of Astronomy as a Science  Michael Tirabassi
Translations from Plato's Dialogues (Loeb edition): The Myth of Er (a multihemispere model of the sky  Republic, X, 616B617). The True Astronomer Must be the Wisest of Men (Epinomis 990a); Pursue Astronomy as you Would Geometry (Republic 530b); Realise
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 Platonic Solids  Paul Bourke
Definitions, formulas, and illustrations of the threedimensional regular polyhedra: tetrahedron, octahedron, hexahedron (cube), icosahedron, dodecahedron, including the solids as drawn in Kepler's Mysterium Cosmographicum, and as represented in stone
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 Plato's Molecule  Ivars Peterson (MathTrek)
In his dialogue Timaeus, the Greek philosopher Plato (427347 B.C.) carefully laid out his reasoning for ascribing certain geometric shapes to the minuscule particles that constituted the four elements of matter. In his view, these elements  fire, air,
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 Playing with RuthAaron Pairs  Ivars Peterson (MathLand)
On April 8, 1974, Henry (Hank) Aaron hit his 715th major league home run, surpassing the previous mark of 714 career home runs long held by baseball great Babe Ruth. Understandably, the event received considerable coverage in newspapers and magazines
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 Playing Your Cards Right  Ivars Peterson; Science News Online
Researchers in Alberta are working to create a computer program that plays championship poker. The current version of the Alberta poker program, named Loki for the Norse god of mischief and chaos, already plays a strong game, but it isn't ready yet to
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 Plus Magazine  Department of Applied Mathematics and Theoretical Physics, University of Cambridge
Plus (formerly PASSMaths) is a triannual online magazine for older pupils and the general public. It publishes articles explaining the diverse applications of mathematics used in solving problems in physics, chemistry, biology, engineering, and economics.
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 Poe's Secret  Ivars Peterson (MathTrek)
Writer Edgar Allan Poe (18091849) is famous for his short stories of the mysterious and the macabre. His popular tale "The GoldBug," published in 1843, is often cited as one of the best works of fiction that turn upon a secret message. Poe had a longstanding
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 Population Index 19861998  Office of Population Research, Princeton University
The primary reference tool to the world's population literature, presenting an annotated bibliography of recently published books, journal articles, working papers, and other material on population topics. It covers all fields of interest to demographers,
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