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1
Content available remote

Polynomial extension of Fleck's congruence

100%
Sun Z.-W.
Acta Arithmetica
|
2006
|
tom 122
|
nr 1
91-100
2
Content available remote

Combinatorial congruences and Stirling numbers

100%
Sun Z.-W.
Acta Arithmetica
|
2007
|
tom 126
|
nr 4
387-398
3
Content available remote

p-adic valuations of some sums of multinomial coefficients

100%
Sun Z.-W.
Acta Arithmetica
|
2011
|
tom 148
|
nr 1
63-76
4
Content available remote

Mixed sums of squares and triangular numbers

100%
Sun Z.-W.
Acta Arithmetica
|
2007
|
tom 127
|
nr 2
103-113
5
Content available remote

Fibonacci numbers and Fermat's last theorem

100%
Sun Z.-W.
Acta Arithmetica
|
1991-1992
|
tom 60
|
nr 4
371-388
EN
Let {Fₙ} be the Fibonacci sequence defined by F₀=0, F₁=1, $F_{n+1}=Fₙ+F_{n-1} (n≥1)$. It is well known that $F_{p-(5/p)}≡ 0 (mod p)$ for any odd prime p, where (-) denotes the Legendre symbol. In 1960 D. D. Wall [13] asked whether $p²|F_{p-(5/p)}$ is always impossible; up to now this is still open. In this paper the sum $∑_{k≡ r (mod 10)}{n\choose k}$ is expressed in terms of Fibonacci numbers. As applications we obtain a new formula for the Fibonacci quotient $F_{p-(5/p)}/p$ and a criterion for the relation $p|F_{(p-1)/4}$ (if p ≡ 1 (mod 4), where p ≠ 5 is an odd prime. We also prove that the affirmative answer to Wall's question implies the first case of FLT (Fermat's last theorem); from this it follows that the first case of FLT holds for those exponents which are (odd) Fibonacci primes or Lucas primes.
6
Content available remote

On sums involving products of three binomial coefficients

100%
Sun Z.-W.
Acta Arithmetica
|
2012
|
tom 156
|
nr 2
123-141
7
Content available remote

On sums of binomial coefficients modulo p²

100%
Sun Z.-W.
Colloquium Mathematicum
|
2012
|
tom 127
|
nr 1
39-54
EN
Let p be an odd prime and let a be a positive integer. In this paper we investigate the sum $∑_{k=0}^{p^{a}-1} (hp^{a}-1 \atop k) (2k \atop k)/m^{k} (mod p²)$, where h and m are p-adic integers with m ≢ 0 (mod p). For example, we show that if h ≢ 0 (mod p) and $p^{a} > 3$, then $∑_{k=0}^{p^{a}-1} (hp^{a}-1 \atop k)(2k \atop k)(-h/2)^{k} ≡ ((1-2h)/(p^{a}))(1 + h((4-2/h)^{p-1} - 1)) (mod p²)$, where (·/·) denotes the Jacobi symbol. Here is another remarkable congruence: If $p^{a} > 3$ then $∑_{k=0}^{p^{a}-1} (p^{a}-1 \atop k)(2k \atop k)(-1)^{k} ≡ 3^{p-1} (p^{a}/3) (mod p²)$.
8
Content available remote

Products of binomial coefficients modulo p²

88%
Sun Z.-W.
Acta Arithmetica
|
2001
|
tom 97
|
nr 1
87-98
9
Content available remote

Restricted sums of subsets of ℤ

88%
Sun Z.-W.
Acta Arithmetica
|
2001
|
tom 99
|
nr 1
41-60
10
Content available remote

Arithmetic theory of harmonic numbers (II)

64%
Sun Z.-W., Zhao L.-L.
Colloquium Mathematicum
|
2013
|
tom 130
|
nr 1
67-78
EN
For k = 1,2,... let $H_k$ denote the harmonic number $∑_{j=1}^k 1/j$. In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime p > 3 we have $∑_{k=1}^{p-1} (H_k)/(k2^k) ≡ 7/24 pB_{p-3} (mod p²)$, $∑_{k=1}^{p-1} (H_{k,2})/(k2^k) ≡ - 3/8 B_{p-3} (mod p)$, and $∑_{k=1}^{p-1} (H²_{k,2n})/(k^{2n}) ≡ (\binom{6n+1}{2n-1} + n)/(6n+1) pB_{p-1-6n} (mod p²)$ for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and $H_{k,m}: = ∑_{j=1}^k 1/(j^m)$.
11
Content available remote

Integers not of the form $c(2^a + 2^b) + p^{α}$

64%
Sun Z.-W., Le M.-H.
Acta Arithmetica
|
2001
|
tom 99
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nr 2
183-190
12
Content available remote

On sums of distinct representatives

64%
Cao H.-Q., Sun Z.-W.
Acta Arithmetica
|
1998-1999
|
tom 87
|
nr 2
159-169
13
Content available remote

On Bialostocki's conjecture for zero-sum sequences

64%
Guo S., Sun Z.-W.
Acta Arithmetica
|
2009
|
tom 140
|
nr 4
329-334
14
Content available remote

On q-Euler numbers, q-Salié numbers and q-Carlitz numbers

64%
Pan H., Sun Z.-W.
Acta Arithmetica
|
2006
|
tom 124
|
nr 1
41-57
15
Content available remote

Restricted sums in a field

64%
Hou Q.-H., Sun Z.-W.
Acta Arithmetica
|
2002
|
tom 102
|
nr 3
239-249
16
Content available remote

Identities concerning Bernoulli and Euler polynomials

64%
Sun Z.-W., Pan H.
Acta Arithmetica
|
2006
|
tom 125
|
nr 1
21-39
17
Content available remote

A characterization of covering equivalence

64%
Pan H., Sun Z.-W.
Acta Arithmetica
|
2007
|
tom 129
|
nr 4
397-402
18
Content available remote

On Fleck quotients

64%
Sun Z.-W., Wan D.
Acta Arithmetica
|
2007
|
tom 127
|
nr 4
337-363
19
Content available remote

On some universal sums of generalized polygonal numbers

64%
Ge F., Sun Z.-W.
Colloquium Mathematicum
|
2016
|
tom 145
|
nr 1
149-155
EN
For m = 3,4,... those pₘ(x) = (m-2)x(x-1)/2 + x with x ∈ ℤ are called generalized m-gonal numbers. Sun (2015) studied for what values of positive integers a,b,c the sum ap₅ + bp₅ + cp₅ is universal over ℤ (i.e., any n ∈ ℕ = {0,1,2,...} has the form ap₅(x) + bp₅(y) + cp₅(z) with x,y,z ∈ ℤ). We prove that p₅ + bp₅ + 3p₅ (b = 1,2,3,4,9) and p₅ + 2p₅ + 6p₅ are universal over ℤ, as conjectured by Sun. Sun also conjectured that any n ∈ ℕ can be written as $p₃(x) + p₅(y) + p_{11}(z)$ and 3p₃(x) + p₅(y) + p₇(z) with x,y,z ∈ ℕ; in contrast, we show that $p₃ + p₅ + p_{11}$ and 3p₃ + p₅ + p₇ are universal over ℤ. Our proofs are essentially elementary and hence suitable for general readers.
20
Content available remote

Some congruences involving binomial coefficients

64%
Cao H.-Q., Sun Z.-W.
Colloquium Mathematicum
|
2015
|
tom 139
|
nr 1
127-136
EN
Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that $T_{p-1} ≡ (p/3) 3^{p-1} (mod p²)$, where the central trinomial coefficient Tₙ is the constant term in the expansion of $(1 + x + x^{-1})ⁿ$. We also prove three congruences modulo p³ conjectured by Sun, one of which is $∑_{k=0}^{p-1} \binom{p-1}{k}\binom{2k}{k} ((-1)^k - (-3)^{-k}) ≡ (p/3)(3^{p-1} - 1) (mod p³)$. In addition, we get some new combinatorial identities.
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