Proof of two conjectures of Guo and Schlosser

The Ramanujan Journal - In this paper, we will investigate three Goldbach–Linnik type problems on cubes of primes. For example, it was proved that, for $$k=1364,$$ every pair of sufficiently...     

Proof of two conjectures of Móricz on double trigonometric series

We use a unified approach to obtain several integrability theorems of Boas [R.P. Boas, Integrability of trigonometric series, I, Duke Math. J. 18 (1951) 787-793]. In particular, we settle two conjectures of Móricz [F. Móricz, On the integrability of double cosine and sine series, II, J. Math. Anal. Appl. 154 (1991) 466-483] concerning double trigonometric series.     

Proof of three conjectures on congruences

This paper proves three conjectures on congruences involving central binomial coefficients or Lucas sequences.Let p be an odd prime and let a be a positive integer.It is shown that if p=1(mod 4)or a1then where(—)denotes the Jacobi symbol.This confirms a conjecture of the second author.A conjecture of Tauraso is also confirmed by showing that where the Lucas numbers Lo,L_1,L_2,...are defined by L_0=2,L_1=1 and L_n+1=L_n+L_n-l(n=1,2,3,...).The third theorem states that if p=5 then F_p~a-(p~a/5)mod p~3 can be determined in the following way:which appeared as a conjecture in a paper of Sun and Tauraso in 2010.     

Proof of some congruence conjectures of Guo and Liu

The Ramanujan Journal - Let n and r be positive integers. Define the numbers $$S_{n}^{(r)}$$ by $$S_{n}^{(r)}=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left(...     

Counterexamples to two conjectures of Hilton

Counterexamples are presented to the following two conjectures of Hilton:

• 1 A graph which does not contain a spanning K_1t is Vl-critical if and only if it is VC-critical.
• 2 If G is a class-two graph which contains a spanning Pl-critical subgraph H of the same chromatic index as G, then G is Vl-critical.

The counterexample to the second conjecture also illustrates that a class-two graph can have distinct Pl-critical subgraphs.     

Essential domains and two conjectures in dimension theory

This note investigates two long-standing conjectures on the Krull dimension of integer-valued polynomial rings and of polynomial rings in the context of (locally) essential domains.

     

关于立方阶数列及其两个猜想

对任意正整数n,设{cn}表示立方数列,即cn=n3.而立方阶数列{Zn}定义为最小的正整数Zn使得czn/n≡l(mod cn 1).本文的主要目的是利用初等森研究数列{zn}的计算问题,并给出了Zn的具体表示形式,从而证明了Kenichiro Kashihara提出的两个猜想:A.数列{zn}中除了第一项外,其余项都是偶数与B.在数列{zn}中存在无限多个平方数是正确的.     

On two conjectures of P. Chowla and S. Chowla concerning continued fractions

Summary The alternating sum of the partial quotients in the primitive period of a continued fraction expansion of √D is determined mod 2 and mod 3. To Professor BeniaminoSegre on his 70-th birthday. Pervenuto il 7 maggio 1973.     

The equivalence of two conjectures of Berge and Fulkerson

Let G be a bridgeless cubic graph. Fulkerson conjectured that there exist six 1‐factors of G such that each edge of G is contained in exactly two of them. Berge conjectured that the edge‐set of G can be covered with at most five 1‐factors. We prove that the two conjectures are equivalent. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:125‐128, 2011     

Integral group rings of frobenius groups and the conjectures of H.J. Zassenhaus

The conjecture of H.J. Zassenhaus for finite subgroups of units of integral group rings. restricted to p-subgroups, is proved for finite Frobenius groups when p is an odd prime. The result for 2-subgroups is established for those Frobenius groups that cannot be mapped homo-morphically onto S$sub:5$esub:. The conjecture in its full strength is proved for A₅, S₅ and SL(2.5).