360 research outputs found
Some congruences involving central q-binomial coefficients
Motivated by recent works of Sun and Tauraso, we prove some variations on the
Green-Krammer identity involving central q-binomial coefficients, such as where is
the Legendre symbol and is the th cyclotomic polynomial. As
consequences, we deduce that \sum_{k=0}^{3^a m-1} q^{k}{2k\brack k}_q
&\equiv 0 \pmod{(1-q^{3^a})/(1-q)}, \sum_{k=0}^{5^a m-1}(-1)^kq^{-{k+1\choose
2}}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{5^a})/(1-q)}, for , the
first one being a partial q-analogue of the Strauss-Shallit-Zagier congruence
modulo powers of 3. Several related conjectures are proposed.Comment: 16 pages, detailed proofs of Theorems 4.1 and 4.3 are added, to
appear in Adv. Appl. Mat
On congruences related to central binomial coefficients
It is known that and
. In this paper we obtain
their p-adic analogues such as
where p>3 is a prime and E_0,E_1,E_2,... are Euler
numbers. Besides these, we also deduce some other congruences related to
central binomial coefficients. In addition, we pose some conjectures one of
which states that for any odd prime p we have
if (p/7)=1 and p=x^2+7y^2
with x,y integers, and if
(p/7)=-1, i.e., p=3,5,6 (mod 7)
In search of Robbins stability
We speculate on whether a certain p-adic stability phenomenon, observed by
David Robbins empirically for Dodgson condensation, appears in other nonlinear
recurrence relations that "unexpectedly" produce integer or nearly-integer
sequences. We exhibit an example (number friezes) where this phenomenon
provably occurs.Comment: 10 pages; to appear in the David Robbins memorial issue of Advances
in Applied Mathematic
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