Search Loci: Convergence:
How can a modern anthropologist embark upon a generalization with any hope of arriving at a satisfactory conclusion? By thinking of the organizational ideas that are present in any society as a mathematical pattern.
Rethinking Anthropology, 1961.
Sums of Powers of Positive Integers
– Apostol, Tom M., 2008, “A Primer on Bernoulli Numbers and Polynomials,” Mathematics Magazine 81:3, 178-190.
– Baron, Margaret E., 1987, The Origins of the Infinitesimal Calculus, New York: Dover; first published by Pergamon Press, Oxford, 1969.
– Beery, Janet, and Jacqueline Stedall, 2009, Thomas Harriot’s Doctrine of Triangular Numbers: the “Magisteria Magna”, Zurich: European Mathematical Society.
– Bernoulli, Jakob, 1713, Ars Conjectandi, Basel.
– Boyer, Carl B., 1943, “Pascal’s Formula for the Sums of Powers of the Integers,” Scripta Mathematica 9, 237-244.
– Briggs, Henry, 1624, Arithmetica Logarithmica, London.
– Edwards, A.W.F., 1987, Pascal’s Arithmetical Triangle, London: Charles Griffin.
– Edwards, C. H., 1980, The Historical Development of the Calculus, New York: Springer.
– Faulhaber, Johannes, 1631, Academia Algebrae, Augsburg; accessed through University of Dresden Digital Library, http://digital.slub-dresden.de/en/sammlungen/titeldaten/272635758/
– Frisch, Christian (editor), 1858-1872, Joannis Kepleri Astronomi Opera Omnia, Vol. IV, Frankfurt.
– Hakewill, George, 1630, An Apologie of the Power and Providence of God in the Government of the World (2nd ed.), Oxford.
– Harriot, Thomas, “De Numeris Triangularibus et inde De Progressionibus Arithmeticis,” British Library Additional MS 6782, ff. 107-146v.
– Harriot, Thomas, “Ad aggregata [Squares], [Cubes], [Square-squares], et c. (For Sums of Squares, Cubes, Square-squares, Etc.),” British Library Additional MS 6782, ff. 239-240v.
– Harriot, Thomas, “Ad Progressiones,” British Library Additional MS 6782, ff. 234-236.
– Heath, Thomas L., 1981, A History of Greek Mathematics, Vol. II, New York: Dover; first published by Clarendon Press, Oxford, 1921.
– Hutton, Charles, 1812, Tracts on Mathematical and Philosophical Subjects, Vol. I, London.
– Katz, Victor, 1998, A History of Mathematics: An Introduction (2nd ed.), Reading, Mass.: Addison-Wesley.
– Knuth, Donald E., 1993, “Johann Faulhaber and Sums of Powers,” Mathematics of Computation 61:203, 277-294.
– Mahoney, Michael, 1994, The Mathematical Career of Pierre de Fermat (2nd ed.), Princeton University Press.
– Maurolico, Francisco, 1575, Arithmeticorum Libri Duo, Venice.
– Nelsen, Roger B., Proofs Without Words, 1993, Washington, DC: Mathematical Association of America.
– Pascal, Blaise, 1665, Traite du Triangle Arithmetique, Paris.
– Recorde, Robert, 1557, Whetstone of Witte, London; reprinted 1969, New York: Da Capo Press.
– Schneider, Ivo, 1993, Johannes Faulhaber (1580-1635): Rechenmeister in einer Welt des Umbruchs, Basel: Birkhauser.
– Smith, David Eugene (editor), 1964, A Source Book in Mathematics, New York: Dover; first published by McGraw-Hill, 1929.
– Stein, Sherman, 1999, Archimedes: What Did He Do Besides Cry Eureka?, Washington, DC: Mathematical Association of America.
– Struik, D. J. (editor), 1969, A Source Book in Mathematics (1200-1800), Cambridge, Mass.: Harvard University Press.
Beery, Janet, "Sums of Powers of Positive Integers," Loci (February 2009), DOI: 10.4169/loci003284
I just wonder if the Karaji Generalized his method of solutions? Is there any information about that. For instance did he found sum of fifth powers?
Al-Karaji and higher powers by Janet Beery (posted: 01/20/2010 )
On the symmetry of the resulting polynomials
== Conjecture == It would be interesting to know that * sums of odd powers can be factored as a polynomial on (n Ã¢â¹â¦ (n + 1) / 2) and is therefore symmetric with the center at (Ã¢ËâÃÂ½); and that * sums of even powers can be factored to a function like above multiplied by (2 Ã¢â¹â¦ n + 1) and therefore antisymmetric with the center at (Ã¢ËâÃÂ½). It would be interesting to know whether any rule concerning the polynomials thus obtained.
Re: On the symmetry of the resulting polynomials by Rick Mabry (posted: 12/30/2011 )
More about symmetry: by Janet Beery (posted: 01/03/2013 )