Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl
Preferencje help
Polish version English version Język
Widoczny [Schowaj] Abstrakt
Liczba wyników

Znaleziono wyników: 12

Liczba wyników na stronie
first rewind previous Strona / 1 next fast forward last

Wyniki wyszukiwania

help Sortuj według:

help Ogranicz wyniki do:
first rewind previous Strona / 1 next fast forward last
1
Content available remote

Supplements to the theory of quartic residues

100%
Sun Z.-H.
Acta Arithmetica
|
2001
|
tom 97
|
nr 4
361-377
2
Content available remote

Binary quadratic forms and sums of triangular numbers

100%
Sun Z.-H.
Acta Arithmetica
|
2011
|
tom 146
|
nr 3
257-297
3
Content available remote

On the theory of cubic residues and nonresidues

100%
Sun Z.-H.
Acta Arithmetica
|
1998
|
tom 84
|
nr 4
291-335
4
Content available remote

On the number of representations of n by ax²+by(y-1)/2, ax²+by(3y-1)/2 and ax(x-1)/2+by(3y-1)/2

100%
Sun Z.-H.
Acta Arithmetica
|
2011
|
tom 147
|
nr 1
81-100
5
Content available remote

Congruences for $(A+√(A²+mB²))^{(p-1)/2)}$ and $(b + √(a²+b²))^{(p-1)/4} (mod p)$

100%
Sun Z.-H.
Acta Arithmetica
|
2011
|
tom 149
|
nr 3
275-296
6
Content available remote

On the quartic character of quadratic units

100%
Sun Z.-H.
Acta Arithmetica
|
2013
|
tom 159
|
nr 1
89-100
EN
Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of integers m and n. Let p be a prime of the form 4k+1 and p = c²+d² with c,d ∈ ℤ, $d=2^r d₀$ and c ≡ d₀ ≡ 1 (mod 4). In the paper we determine $(b+√(b²+4^α)/2)^{(p-1)/4)} (mod p)$ for p = x²+(b²+4^{α})y² (b,x,y ∈ ℤ, 2∤b), and $(2a+√{4a²+1})^{(p-1)/4} (mod p)$ for p = x²+(4a²+1)y² (a,x,y∈ℤ) on the condition that (c,x+d) = 1 or (d₀,x+c) = 1. As applications we obtain the congruence for $U_{(p-1)/4} (mod p)$ and the criterion for $p | U_{(p-1)/8}$ (if p ≡ 1 (mod 8)), where {Uₙ} is the Lucas sequence given by U₀ = 0, U₁ = 1 and $U_{n+1} = bUₙ+U_{n-1} (n≥1)$, and b ≢ 2 (mod 4). Hence we partially solve some conjectures that we posed in 2009.
7
Content available remote

Congruences for $q^{[p/8]}(mod p)$

100%
Sun Z.-H.
Acta Arithmetica
|
2013
|
tom 159
|
nr 1
1-25
EN
Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of the integers m and n. Let p ≡ 1 (mod 4) be a prime, q ∈ ℤ, 2 ∤ q and p=c²+d²=x²+qy² with c,d,x,y ∈ ℤ and c ≡ 1 (mod 4). Suppose that (c,x+d)=1 or (d,x+c) is a power of 2. In this paper, by using the quartic reciprocity law, we determine $q^{[p/8]}(mod p)$ in terms of c,d,x and y, where [·] is the greatest integer function. Hence we partially solve some conjectures posed in our previous two papers.
8
Content available remote

Legendre polynomials and supercongruences

100%
Sun Z.-H.
Acta Arithmetica
|
2013
|
tom 159
|
nr 2
169-200
EN
Let p > 3 be a prime, and let Rₚ be the set of rational numbers whose denominator is not divisible by p. Let {Pₙ(x)} be the Legendre polynomials. In this paper we mainly show that for m,n,t ∈ Rₚ with m≢ 0 (mod p), $P_{[p/6]}(t) ≡ -(3/p)∑_{x=0}^{p-1} ((x³-3x+2t)/p) (mod p)$ and $(∑_{x=0}^{p-1} ((x³+mx+n)/p))² ≡ ((-3m)/p) ∑_{k=0}^{[p/6]} \binom{2k}{k}\binom{3k}{k}\binom{6k}{3k} ((4m³+27n²)/(12³·4m³))^k (mod p)$, where (a/p) is the Legendre symbol and [x] is the greatest integer function. As an application we solve some conjectures of Z. W. Sun and the author concerning $∑_{k=0}^{p-1}\binom{2k}{k}\binom{3k}{k}\binom{6k}{3k}/m^{k} (mod p²)$, where m is an integer not divisible by p.
9
Content available remote

The expansion of $∏_{k=1}^{∞}(1-q^{ak})(1-q^{bk})$

100%
Sun Z.-H.
Acta Arithmetica
|
2008
|
tom 134
|
nr 1
11-29
10
Content available remote

On the number of representations of n by ax² + bxy + cy²

64%
Sun Z.-H., Williams K.
Acta Arithmetica
|
2006
|
tom 122
|
nr 2
101-171
11
Content available remote

Ramanujan identities and Euler products for a type of Dirichlet series

64%
Sun Z.-H., Williams K.
Acta Arithmetica
|
2006
|
tom 122
|
nr 4
349-393
12
Content available remote

Turán's problem and Ramsey numbers for trees

52%
Sun Z.-H., Wang L.-L., Wu Y.-L.
Colloquium Mathematicum
|
2015
|
tom 139
|
nr 2
273-298
EN
Let T¹ₙ = (V,E₁) and T²ₙ = (V,E₂) be the trees on n vertices with $V = {v₀,v₁,...,v_{n-1}}$, $E₁ = {v₀v₁,..., v₀v_{n-3},v_{n-4}v_{n-2},v_{n-3}v_{n-1}}$ and $E₂ = {v₀v₁,..., v₀v_{n-3},v_{n-3}v_{n-2},v_{n-3}v_{n-1}}$. For p ≥ n ≥ 5 we obtain explicit formulas for ex(p;T¹ₙ) and ex(p;T²ₙ), where ex(p;L) denotes the maximal number of edges in a graph of order p not containing L as a subgraph. Let r(G₁,G₂) be the Ramsey number of the two graphs G₁ and G₂. We also obtain some explicit formulas for $r(Tₘ,Tₙ^i)$, where i ∈ {1,2} and Tₘ is a tree on m vertices with Δ(Tₘ) ≤ m - 3.
first rewind previous Strona / 1 next fast forward last
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.