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1
Content available remote On the congruence f(x) + g(y) + c ≡ 0 (mod xy)
100%
Schinzel A.
Banach Center Publications
|
2016
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tom 108
|
nr 1
255-257
EN
Four theorems of the author on the subject are given without proofs.
2
Content available remote On values of the Mahler measure in a quadratic field (solution of a problem of Dixon and Dubickas)
100%
Schinzel A.
Acta Arithmetica
|
2004
|
tom 113
|
nr 4
401-408
3
Content available remote Corrigendum to the paper "On the greatest common divisor of two univariate polynomials, II" (Acta Arith. 98 (2001), 95-106)
100%
Schinzel A.
Acta Arithmetica
|
2004
|
tom 115
|
nr 4
403
4
Content available remote Stern Polynomials as Numerators of Continued Fractions
100%
Schinzel A.
|
|
tom 62
|
nr 1
23-27
EN
It is proved that the nth Stern polynomial Bₙ(t) in the sense of Klavžar, Milutinović and Petr [Adv. Appl. Math. 39 (2007)] is the numerator of a continued fraction of n terms. This generalizes a result of Graham, Knuth and Patashnik concerning the Stern sequence Bₙ(1). As an application, the degree of Bₙ(t) is expressed in terms of the binary expansion of n.
5
Content available remote Solution to a Problem of Lubelski and an Improvement of a Theorem of His
100%
Schinzel A.
|
|
tom 59
|
nr 2
115-119
EN
The paper consists of two parts, both related to problems of Lubelski, but unrelated otherwise. Theorem 1 enumerates for a = 1,2 the finitely many positive integers D such that every odd positive integer L that divides x² +Dy² for (x,y) = 1 has the property that either L or $2^{a}L$ is properly represented by x²+Dy². Theorem 2 asserts the following property of finite extensions k of ℚ : if a polynomial f ∈ k[x] for almost all prime ideals 𝔭 of k has modulo 𝔭 at least v linear factors, counting multiplicities, then either f is divisible by a product of v+1 factors from k[x]∖ k, or f is a product of v linear factors from k[x].
6
Content available remote On the diophantine equation x²+x+1 = yz
100%
Schinzel A.
|
|
tom 141
|
nr 2
243-248
EN
All solutions of the equation x²+x+1 = yz in non-negative integers x,y,z are given in terms of an arithmetic continued fraction.
7
Content available remote Primitive roots and quadratic non-residues
100%
Schinzel A.
Acta Arithmetica
|
2011
|
tom 149
|
nr 2
161-170
8
Content available remote On sums of powers of the positive integers
100%
Schinzel A.
Colloquium Mathematicum
|
2013
|
tom 132
|
nr 2
211-220
EN
The pairs (k,m) are studied such that for every positive integer n we have $1^{k} + 2^{k} + ⋯ + n^{k} | 1^{km} + 2^{km} + ⋯ + n^{km}$.
9
Content available remote The reduced length of a polynomial with complex coefficients
100%
Schinzel A.
Acta Arithmetica
|
2008
|
tom 133
|
nr 1
73-81
10
Content available remote On Ternary Integral Recurrences
100%
Schinzel A.
|
|
tom 63
|
nr 1
19-23
EN
We prove that if a,b,c,d,e,m are integers, m > 0 and (m,ac) = 1, then there exist infinitely many positive integers n such that m|(an+b)cⁿ - deⁿ. Hence we derive a similar conclusion for ternary integral recurrences.
11
Content available remote Reciprocal Stern Polynomials
100%
Schinzel A.
|
|
tom 63
|
nr 2
141-147
EN
A partial answer is given to a problem of Ulas (2011), asking when the nth Stern polynomial is reciprocal.
12
Content available remote On the congruence f(x) + g(y) + c ≡ 0 (mod xy) (completion of Mordell's proof)
100%
Schinzel A.
|
|
nr 4
347-374
EN
Assertions on the congruence f(x) + g(y) + c ≡ 0 (mod xy) made without proof by Mordell in his paper in Acta Math. 88 (1952) are either proved or disproved.
13
Content available remote A property of the unitary convolution
100%
Schinzel A.
Colloquium Mathematicum
|
1998
|
tom 78
|
nr 1
93-96
14
Content available remote Jerzy Browkin (1934-2015)
88%
Schinzel A.
|
|
tom 172
|
nr 3
199-206
15
Content available remote Corrigendum to the papers "On two theorems of Gelfond and some of their applications" and "On primitive prime factors of Lehmer numbers III"
75%
Schinzel A.
|
|
nr 1
101-102
16
Content available On the composite Lehmer numbers with prime indices, I
75%
Schinzel A.
|
|
tom 9
|
nr 1
EN
The article contains no abstract
17
Content available remote Corrigendum to "Representatons of multivariate polynomials by sums of univariate polynomials in linear forms" (Colloq. Math. 112 (2008), 201-233)
63%
Białynicki-Birula A. , Schinzel A.
Colloquium Mathematicum
|
2011
|
tom 125
|
nr 1
18
Content available remote Comparison of L¹- and $L^{∞}$-norms of squares of polynomials
63%
Schinzel A. , Schmidt W.
|
|
tom 104
|
nr 3
283-296
19
Content available remote On integers not of the form n - φ (n)
63%
Browkin J. , Schinzel A.
Colloquium Mathematicum
|
1995
|
tom 68
|
nr 1
55-58
EN
W. Sierpiński asked in 1959 (see [4], pp. 200-201, cf. [2]) whether there exist infinitely many positive integers not of the form n - φ(n), where φ is the Euler function. We answer this question in the affirmative by proving Theorem. None of the numbers $2^k·509203$ (k = 1, 2,...) is of the form n - φ(n).
20
Content available remote On Equations y² = xⁿ+k in a Finite Field
63%
Schinzel A. , Skałba M.
|
|
nr 3
223-226
EN
Solutions of the equations y² = xⁿ+k (n = 3,4) in a finite field are given almost explicitly in terms of k.
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