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Alexandru Zaharescu ¹, Mohammad Zaki ¹

Affiliations
1 Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL-61801, US

Abstract

In this paper we state and prove an analog of the abc conjecture in the ring A_r (K) of arithmetical functions in r variables over a field K of characteristic zero.

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Marius Cavachi ¹, Marian Vajaitu ², Alexandru Zaharescu ³

Affiliations
1 “Ovidius” University of Constanta, Bd. Mamaia 124, 8700, Constanta, RO
2 Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest 70700, RO
3 Department of Mathematics, University of Illinois at Urbana-Champaign, Altgeld Hall, 1409 W. Green Street, Urbana, IL, 61801, US

Abstract

For any polynomial

f (X) ∈ Q[X], write f (X) in the form f(x) =a₀+a₁X+....+a_dX^d / q

 

with q,a₀,a₁,......,aa_d ∈ Z,q > 1,q as small as possible, then set

           H(f) = max{|a₀|,|a₁|,......,|a_d|,q}

we show that for any relatively prime polynomials f(x), g(x) ∈ Q[X] with deg f < deg g = d, and any prime number p>2d^d H(f)^d+1 H(g)^3d, the polynomial f(X) + pg(X) is irreducible over Q. we also condider the more general case of polynomial defined over a number field.

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