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Sums of the first n odd integers

Published online by Cambridge University Press:  16 February 2023

Chris Sangwin
Chris Sangwin*
Affiliation:
School of Mathematics, University of Edinburgh, EH9 3FD e-mail: C.J.Sangwin@ed.ac.uk
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Extract

There is a long tradition of mathematicians valuing collections of proofs, e.g. [1] is a famous recent collection. In the past, professional mathematicians have also written many different proofs of single important theorems, e.g. [2]. Proofs of the Pythagorean Theorem were collected by [3], and [4] discusses mathematical style through proofs of the irrationality of . The recent work of [5] is a discussion of mathematical style via a comparison of 99 different ‘proofs’ the following ‘theorem’.

Type
Articles
Information
The Mathematical Gazette , Volume 107 , Issue 568 , March 2023 , pp. 10 - 24
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Copyright
© The Authors, 2023. Published by Cambridge University Press on behalf of The Mathematical Association

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References

Aigner, M. and Ziegler, G., Proofs from THE BOOK (6th edn.), Springer (2018).10.1007/978-3-662-57265-8CrossRefGoogle Scholar
Kalman, D., Six ways to sum a series The College Mathematics Journal, 24(5): (1993) pp. 402–421.CrossRefGoogle Scholar
Loomis, E., The pythagorean proposition, National Council of Teachers of Mathematics, Washington (1968).Google Scholar
Duchêne, L. and LeBlanc, A., Rationnel mon Q, Hermann (2010).Google Scholar
Ording, P., 99 Variations on a proof, Princeton University Press, (2019).CrossRefGoogle Scholar
Mejía-Ramos, J. P., Evans, T., Rittberg, C. and Inglis, M., Mathematicians’ assessments of the explanatory value of proofs, Axiomathes, 3 (2021) pp. 1–25.Google Scholar
Pólya, G., Mathematics and plausible reasoning, Vol.1: Induction and analogy in mathematics, Vol.2: Patterns of plausible inference, Princeton University Press (1954).Google Scholar
Edwards, A. W. F., Sums of powers of integers: a little of the history, Math. Gaz. 66 (March 1982) pp. 22–28.CrossRefGoogle Scholar
Burn, R. P., Sums of powers of integers—how Fermat may have found them, Math. Gaz. 94 (March 2010) pp. 18–26.CrossRefGoogle Scholar
Murphy, R. V. W., Summing powers of natural numbers, Math. Gaz. 93 (July 2009) pp. 279–285.CrossRefGoogle Scholar
Howard, F. T., Sums of powers of integers via generating functions, The Fibonacci Quarterly, 34(3) (1996) pp. 244–256.Google Scholar
Kalman, D., Sums of powers by matrix methods, The Fibonacci Quarterly, 28(1) (1990) pp. 60–71.Google Scholar
Cajori, F., Origin of the name ‘mathematical induction’, Amer. Math. Monthly, 25 (1918) pp. 197–201.Google Scholar
De Morgan, A., Mathematical induction, The Penny Cyclopedia XII (1838) pp. 465–466.Google Scholar
Rotman, J. J., Journey into mathematics: an introduction to proofs, Prentice Hall, (1998).Google Scholar
Velleman, D. J., How to prove it: a structured approach, Cambridge University Press (2006).CrossRefGoogle Scholar
Pólya, G., Mathematical discovery: on understanding, learning, and teaching problem solving, Wiley (1962).Google Scholar
Raman-Sundström, M. and Ohman, L., Mathematical fit: a case study, Philosophia Mathematica, 26(2) (2016) pp. 184–210.Google Scholar
Oughtred, W., Mathematical recreations, T. Cotes, London (1633).Google Scholar
Brown, J. R., Philosophy of mathematics: a contemporary introduction to the world of proofs and pictures, Routledge (2008).Google Scholar
D’Ooge, M. L., Nicomachus of Gerasa: introduction to arithmetic, MacMillan (1926).Google Scholar
Nelson, R. B., Proofs without words: exercises in visual thinking, The Mathematical Association of America (1993).Google Scholar
van Schooten, Franciscus, Exercitationes mathematicae libri quinque, Lugd. Batavor.: Ex officina elzeviriorum (1657).Google Scholar
Euler, L., Elements of algebra (3rd edn.), Longman, Hurst, Rees, Orme and Co. (1822). Translated by J. Hewlett from the French, with the notes of M. Bernoulli and the additions of M. de La Grange.Google Scholar
Day, J., An introduction to algebra, being the first part of a course of mathematics (32nd edn.), Durrie & Peck (1838).Google Scholar
Sigler, L. E., Fibonacci’s Liber Abaci: Leonardo Pisaro’s book of calculation, Springer (2002).10.1007/978-1-4613-0079-3CrossRefGoogle Scholar
Hayes, B., Gauss’s day of reckoning, American Scientist 94(3) (2006).CrossRefGoogle Scholar
Tuckey, C. O. and Armistead, W., Algebra (3rd edn), Longmans, Green & Co. (1962).Google Scholar
Boole, G., A Treatise on the calculus of finite differences, Cosimo (2007).Google Scholar
Kinnear, G. and Sangwin, C. J., Contradiction and contrapositive, what is the difference? Scottish Mathematics Council Journal 48 (2018) pp. 84–90.Google Scholar
Todhunter, I., Algebra for the use of colleges and schools (5th edn), MacMillian (1897).Google Scholar
Lund, T., The elements of algebra designed for the use of students in the university (14th edn), Longman, Brown, Green and Longmans (1852).Google Scholar
Euler, L., Introduction to Analysis of the Infinite, volume I, Springer, (1988). Translated by J. Blanton from the Latin Introductio in analysin infinitorum (1748).Google Scholar
Cundy, H. M. and Rollett, A. P., Mathematical Models, Oxford University Press (1961).Google Scholar