3,861 research outputs found
Supercongruences involving dual sequences
In this paper we study some sophisticated supercongruences involving dual
sequences. For define and For any odd
prime and -adic integer , we determine
and modulo
; for example, we establish the new -adic congruence
where denotes the least nonnegative integer with
. For any prime and -adic integer , we determine
modulo (or if ),
and show that We also pose
several related conjectures.Comment: 35 pages, final published versio
New observations on primitive roots modulo primes
We make many new observations on primitive roots modulo primes. For an odd
prime and an integer , we establish a theorem concerning
, where runs over all the primitive roots modulo
among , and denotes the Legendre symbol. On the
basis of our numerical computations, we formulate 35 conjectures involving
primitive roots modulo primes. For example, we conjecture that for any prime
there is a primitive root modulo with a square, and that
for any prime there is a prime with the Bernoulli number
a primitive root modulo . We also make related observations on quadratic
nonresidues modulo primes and primitive prime divisors of some combinatorial
sequences. For example, based on heuristic arguments we conjecture that for any
prime there exists a Fibonacci number which is a quadratic
nonresidue modulo ; this implies that there is a deterministic polynomial
time algorithm to find square roots of quadratic residues modulo a prime .Comment: 23 page
A result similar to Lagrange's theorem
Generalized octagonal numbers are those with . In this paper we mainly show that every positive integer can be written as
the sum of four generalized octagonal numbers one of which is odd. This result
is similar to Lagrange's theorem on sums of four squares. Moreover, for
triples with (including and ),
we prove that any nonnegative integer can be exprssed as
with . We also pose
several conjectures for further research.Comment: 21 pages, final published versio
Quadratic residues and related permutations and identities
Let be an odd prime. In this paper we investigate quadratic residues
modulo and related permutations, congruences and identities. If
are all the quadratic residues modulo among
, then the list (with
the least nonnegative residue of modulo ) is a permutation of
, and we show that the sign of this permutation is
or according as or ,
where is the class number of the imaginary quadratic field . To achieve this, we evaluate the product via Dirichlet's class number
formula and Galois theory. We also obtain some new identities for the sine and
cosine functions; for example, we determine the exact value of for any with
.Comment: 36 pages, final published versio
An additive theorem and restricted sumsets
Let G be any additive abelian group with cyclic torsion subgroup, and let A,
B and C be finite subsets of G with cardinality n>0. We show that there is a
numbering {a_i}_{i=1}^n of the elements of A, a numbering {b_i}_{i=1}^n of the
elements of B and a numbering {c_i}_{i=1}^n of the elements of C, such that all
the sums a_i+b_i+c_i (i=1,...,n) are distinct. Consequently, each subcube of
the Latin cube formed by the Cayley addition table of Z/NZ contains a Latin
transversal. This additive theorem can be further extended via restricted
sumsets in a field
Some new inequalities for primes
For n=1,2,3,... let p_n be the n-th prime. We mainly show that
p_n>n+sum_{k=1}^n p_k/k for all n>124, and sum_{k=1}^n kp_k<n^2p_n/3 for all
n>30.Comment: 6 pages. Make Theorem 1.2 genera
On Motzkin numbers and central trinomial coefficients
The Motzkin numbers
and the central trinomial coefficients
( given by the constant term of have many
combinatorial interpretations. In this paper we establish the following
surprising arithmetic properties of them with any positive integer:
and
also Comment: 21 page
On sums of primes and triangular numbers
We study whether sufficiently large integers can be written in the form
cp+T_x, where p is either zero or a prime congruent to r mod d, and
T_x=x(x+1)/2 is a triangular number. We also investigate whether there are
infinitely many positive integers not of the form (2^ap-r)/m+T_x with p a prime
and x an integer. Besides two theorems, the paper also contains several
conjectures together with related analysis and numerical data. One of our
conjectures states that each natural number not equal to 216 can be written in
the form p+T_x with x an integer and p a prime or zero; another conjecture
asserts that any odd integer n>3 can be written in the form p+x(x+1) with p a
prime and x a positive integer
Products and sums divisible by central binomial coefficients
In this paper we initiate the study of products and sums divisible by central
binomial coefficients. We show that 2(2n+1)binom(2n,n)| binom(6n,3n)binom(3n,n)
for every n=1,2,3,... Also, for any nonnegative integers and we have
and
where
denotes the Catalan number .
Applying this result we obtain two sums divisible by central binomial
coefficients.Comment: 15 pages. Submitted version. Add some conjectures and references
The least modulus for which consecutive polynomial values are distinct
Let and be relatively prime integers. We show that for
any sufficiently large integer (in particular suffices for ), the smallest prime with is
the least positive integer with pairwise
distinct modulo , where is the radical of . We also conjecture
that for any integer the least positive integer such that
$|\{k(k-1)/2\ \mbox{mod}\ m:\ k=1,\ldots,n\}|= |\{k(k-1)/2\ \mbox{mod}\ m+2:\
k=1,\ldots,n\}|=np\ge 2n-1p+2$ also prime.Comment: Final published versio
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