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1
Content available remote

On the greatest common divisor of two univariate polynomials, II

100%
Schinzel A.
Acta Arithmetica
|
2001
|
tom 98
|
nr 1
95-106
2
Content available remote

Reciprocal Stern Polynomials

100%
Schinzel A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2015
|
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.
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.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2014
|
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

On Ternary Integral Recurrences

100%
Schinzel A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2015
|
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.
6
Content available remote

Extensions of Three Theorems of Nagell

100%
Schinzel A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2013
|
tom 61
|
nr 3
195-200
EN
Three theorems of Nagell of 1923 concerning integer values of certain sums of fractions are extended.
7
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Solution to a Problem of Lubelski and an Improvement of a Theorem of His

100%
Schinzel A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2011
|
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].
8
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Note to the Paper "A Positive Definite Binary Quadratic Form as a Sum of Five Squares of Linear Forms (Completion of Mordell's Proof)" (Bull. Polish Acad. Sci. Math. 61 (2013), 23-26)

100%
Schinzel A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2013
|
tom 61
|
nr 2
107
9
Content available remote

Reducibility of lacunary polynomials XII

100%
Schinzel A.
Acta Arithmetica
|
1999
|
tom 90
|
nr 3
273-289
10
Content available remote

On Sums of Four Coprime Squares

100%
Schinzel A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2013
|
tom 61
|
nr 2
109-111
EN
It is proved that all sufficiently large integers satisfying the necessary congruence conditions mod 24 are sums of four squares prime in pairs.
11
Content available remote

A Positive Definite Binary Quadratic Form as a Sum of Five Squares of Linear Forms (Completion of Mordell's Proof)

100%
Schinzel A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2013
|
tom 61
|
nr 1
23-26
EN
The paper completes an incomplete proof given by L. J. Mordell in 1930 of the following theorem: every positive definite classical binary quadratic form is the sum of five squares of linear forms with integral coefficients.
12
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 $1^{k} + 2^{k} + ⋯ + n^{k} | 1^{km} + 2^{km} + ⋯ + n^{km}$.
13
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On the congruence f(x) + g(y) + c ≡ 0 (mod xy) (completion of Mordell's proof)

100%
Schinzel A.
Acta Arithmetica
|
2015
|
tom 167
|
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.
14
Content available remote

Primitive roots and quadratic non-residues

100%
Schinzel A.
Acta Arithmetica
|
2011
|
tom 149
|
nr 2
161-170
15
Content available remote

On the Mahler measure of polynomials in many variables

100%
Schinzel A.
Acta Arithmetica
|
1997
|
tom 79
|
nr 1
77-81
16
Content available remote

Reducibility of Symmetric Polynomials

100%
Schinzel A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2005
|
tom 53
|
nr 3
251-258
EN
A necessary and sufficient condition is given for reducibility of a symmetric polynomial whose number of variables is large in comparison to degree.
17
Content available remote

On the diophantine equation x²+x+1 = yz

100%
Schinzel A.
Colloquium Mathematicum
|
2015
|
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.
18
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
Content available

On the composite Lehmer numbers with prime indices, I

100%
Schinzel A.
Commentationes Mathematicae
|
1965
|
tom 9
|
nr 1
EN
The article contains no abstract
19
Content available remote

On the congruence f(x) + g(y) + c ≡ 0 (mod xy)

100%
Schinzel A.
Banach Center Publications
|
2016
|
tom 108
|
nr 1
255-257
EN
Four theorems of the author on the subject are given without proofs.
20
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
Content available

O równaniu \(x^4 + ax^2 y^2 + by^4 = z^{2}\)

100%
Schinzel A.
Commentationes Mathematicae
|
1960
|
tom 4
|
nr 1
EN
The article contains no abstract
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