Atlas: Generalization of Bunyakovsky Theorem in Case of Complicated Radicals and Its Application to Diophantine Equations by Mikhail Novikov

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Turku Symposium on Number Theory in Memory of Kustaa Inkeri
May 31 - June 4, 1999
University of Turku
Turku, Finland

Organizers
Matti Jutila, Tauno Metsänkylä

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Generalization of Bunyakovsky Theorem in Case of Complicated Radicals and Its Application to Diophantine Equations
by
Mikhail Novikov
St. Petersburg Technical University, Institute of Precise Mechanics and Optics

Bunyakovsky generalized theorem in the form of the inequality

n
å
i=1
\alpha_i a_i^1/m_i =/= 0 ,
(1)
where the \alpha_i are rationals and the number a_i, m_i are positive integers, is proved. The arithmetic of the Bunyakovsky polynomials

B(a₁^1/m₁, ... , a_n^1/m_n)= n
å
i=1
\alpha_i·a_i^1/m_i
(2)
as a generalization of Besicovitch polynomials [K. Chandrasekharan, Arithmetical Functions, p. 204] is considered. The Bunyakovsky theorem for composite radicals in the form of the inequality

n
å
i=1
R_i =/= 0,

where the R_i are composite radicals, is proved.

Appplications of the formulas (1)- (3) to Diophantine equations are shown.

Date received: April 7, 1999