Abstract:Abstract. Let x, y and z be positive integers such that x =y + z and gcd (x,y,z)= = 1. We give upper and lower bounds for x in terms of the greatest squarefree divisor of xyz.
We show that there exists an infinite sequence of sums P: a+b=c of rational integers with large height compared to the radical:-2?Âe>1.517 for l=0.5990. This improves the result of Stewart and Tijdeman [9]. The value of l comes from an asymptotic bound for the packing density of spheres. We formulate our result such that improved knowledge of l immediately improves the value of K l .
Academic PressKey Words: abc conjecture; good abc examples; the error term in the abc conjecture; sphere packing in n dimensions.
We show that there exists an infinite sequence of sums P: a+b=c of rational integers with large height compared to the radical:-2?Âe>1.517 for l=0.5990. This improves the result of Stewart and Tijdeman [9]. The value of l comes from an asymptotic bound for the packing density of spheres. We formulate our result such that improved knowledge of l immediately improves the value of K l .
Academic PressKey Words: abc conjecture; good abc examples; the error term in the abc conjecture; sphere packing in n dimensions.
“…By means of the theory of logarithmic forms Stewart and Tijdeman [43] and Stewart and Yu [44], [45] obtained upper bounds for c as a function of N (a, b, c). The best known upper bounds are due to Stewart and Yu [45].…”
mentioning
confidence: 99%
“…The conjecture has many profound consequences; cf. [1], [6], [9], [13], [16], [19], [28], [29], [34], [35], [37], [43], [47], [48] and the references given there.…”
On the abc conjecture in algebraic number fields by K. Győry (Debrecen)To Professor W. M. Schmidt on his 75th birthdayIn our paper we obtain new, effective results (cf. Theorems 1 and 2 in Section 3) towards the truth of the uniform abc conjecture in number fields. Our theorems improve upon the earlier estimates established in this direction. Our Theorems 1 and 2 are deduced from some recent explicit results of Yu and the author [24] (cf. Theorem A in Section 2) concerning S-unit equations. Our proofs (cf. Section 4) depend ultimately on the best known estimates for linear forms in logarithms of algebraic numbers.
“…An unconditional result in the direction of the abc conjecture has been obtained in 1986 by Stewart and Tijdeman [66] using lower bounds for linear combinations of logarithms, in the complex case as well as in the p-adic case: log c Ä ÄR 15 with an absolute constant Ä. This estimate has been refined by Stewart and Yukunrui, who proved in 1991 [67]: for any " > 0 and for c sufficiently large in terms of ",…”
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