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The Schur Derivative of a Polynomial
Published online by Cambridge University Press: 18 May 2009
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For a given sequence {am} and p≠0, Schur (2) defined
In particular if p is a prime, a an integer and 
, then by Fermat's theorem
is integral. Schur proved that if p † a, then all the derivatives
are integral. Zorn (3) using p-adic methods proved Schur's results and also found the residue of Xm (mod pm), where 
and x = 1 (mod p). The writer (1) proved Zorn's congruences by elementary methods as well as certain additional results of a similar sort.
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- Research Article
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- Copyright © Glasgow Mathematical Journal Trust 1953
References
REFERENCES
(1)Carlitz., L., “Some theorems on the Schur derivative,” Pacific Journal of Mathematics, vol. 3 (1953), pp. 321–332.CrossRefGoogle Scholar
(2)Schur, I., “Ein Beitrag zur elementaren Zahlentheorie,” Sitzungaberichte der Preussischen Akademie der Wissenschaften (1933), pp. 145–151.Google Scholar
(3)Zorn, M., “p-adic analysis and elementary number theory,” Annals of Mathematics (2) vol. 38 (1937), pp. 451–464.CrossRefGoogle Scholar
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