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A Generalization of Circulants
Published online by Cambridge University Press: 20 January 2009
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In connection with Bertrand's algebraical exercise of 1850, Muir remarks that it is not unlikely that the divisibility of a3 + b3 + c3 − 3abc by a + b + c had been previously noted, although. there is no record of the fact. Bertrand's exercise is to the effect that the circulant of the third order repeats under multiplication or, what is the same, admits of composition; the formulae of composition are stated in the exercise precisely as they would follow from Spottiswoode's theorem on the linear factors of a circulant. The whole of this is implied as an immediate special case, and indeed as one that any reader would construct at once, in an identity due to Lagrange, reproduced by Legendre.
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- Copyright © Edinburgh Mathematical Society 1928
References
1 History of the Theory of Determinants, vol. 1, p. 401.Google Scholar
2 Theorie des Nombres, tome II, § XVI, pp. 137–138. Bertrand's exercise is obtained from Legendre's formulas by putting a = b = 0, c = 1. The substance of the discussion given also in the Supplements to Euler's Algebra.Google Scholar
3 See Muir, , vol. 4, p. 360. A short account of Schapira's extensive theory will be found in his obituary notices. The original is in Russian.Google Scholar
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