In 1997 Ken Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that
under the generalized Riemann hypothesis any positive odd integer greater than
2719 can be represented by the famous Ramanujan form $x^2+y^2+10z^2$,
equivalently the form $2x^2+5y^2+4T_z$ represents all integers greater than
1359, where $T_z$ denotes the triangular number $z(z+1)/2$.

In this paper we establish some sophisticated congruences involving central
binomial coefficients and Fibonacci numbers. For example, we show that if
$p\not=2,5$ is a prime then
$$\sum_{k=0}^{p-1}F_{2k}\binom{2k}{k}=(-1)^{[p/5]}(1-(p/5)) (mod p^2)$$ and
$$\sum_{k=0}^{p-1}F_{2k+1}\binom{2k}k=(-1)^{[p/5]}(p/5) (mod p^2).$$ We also
obtain similar results for some other second-order recurrences.

We collect here various conjectures on congruences made by the author in a
series of papers, some of which involve binary quadratic forms and other
advanced theories. Part A consists of unsolved conjectures of the author while
conjectures in Part B have been recently confirmed. We hope that this material
will interest number theorists and stimulate further research. Number theorists
are welcome to work on those open conjectures.

In this paper we obtain several congruences modulo an odd prime p which are
related to central binomial coefficients. For example,
$$\sum_{k=0}^{p-1}\binom{2k}{k}^3/64^k=\cases4x^2 (mod p)&if p=x^2+y^2 with x
odd and y even, \\0 (mod p)& if p=3 (mod 4);$$ and
$$\sum_{k=0}^{p-1}C_k^2}/16^k=-3 (mod p) and \sum_{k=0}^{p-1}C_k^3/64^k=7 (mod
p),$$ where $C_k$ denotes the Catalan number $\binom{2k}{k}/(k+1)$. We also
pose several challenging conjectures one of which states that if p=3,5,6 (mod
7)$ then $$\sum_{k=0}^{p-1}\binom{2k}{k}^3=0 (mod p^2).$$

Sun and Tauraso conjectured that for any positive integer $a$ we have
$$\sum_{k=0}^{3^a-1}\binom{2k}{k}=0 (mod 3^{2a})$$ and furthermore
$$3^{-2a}}\sum_{k=0}^{3^a-1}\binom{2k}k=1 (mod 3).$$ Recently a $q$-analogue of
the first congruence was conjectured by Guo and Zeng. In this paper we prove
both conjectures.

Let $m$ and $n>0$ be integers. Suppose that $p$ is a prime dividing $m-4$ but
not dividing $m$. We show that
$$\nu_p(\sum_{k=0}^{n-1}\frac{\binom{2k}k}{m^k})$ is at least $\nu_p(n)$, where
$\nu_p(x)$ denotes the $p$-adic valuation of $x$ at $p$. Furthermore, if
$p\not=3$ or $3\nmid n$ then
$$n^{-1}\sum_{k=0}^{n-1}\frac{\bi{2k}k}{m^k}=\frac{\binom{2n-1}{n-1}}{m^{n-1}}
(mod p^{\nu_p(m-4)}).$$ This implies several conjectures of Guo and Zeng [GZ].

Let p be a prime and let a be a positive integer. In this paper we determine
$\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ and
$\sum_{k=1}^{p-1}\binom{2k}{k+d}/(km^{k-1})$ modulo $p$ for all d=0,...,p^a,
where m is any integer not divisible by p. For example, we show that if
$p\not=2,5$ then
$$\sum_{k=1}^{p-1}(-1)^k\frac{\binom{2k}k}k=-5\frac{F_{p-(\frac p5)}}p (mod
p),$$ where F_n is the n-th Fibonacci number and (-) is the Jacobi symbol. We
also prove that if p>3 then $$\sum_{k=1}^{p-1}\frac{\binom{2k}k}k={8/9}
p^2B_{p-3} (mod p^3),$$ where B_n denotes the n-th Bernoulli number.

We prove that for any nonnegative integers $n$ and $r$ the binomial sum $$
\sum_{k=-n}^n\binom{2n}{n-k}k^{2r} $$ is divisible by
$2^{2n-\min\{\alpha(n),\alpha(r)\}}$, where $\alpha(n)$ denotes the number of
1's in the binary expansion of $n$. This confirms a recent conjecture of Guo
and Zeng.

Let p be any odd prime. We mainly show that
$$\sum_{k=1}^{p-1}binomial(3k,k)*2^k/k=0 (mod p)$$ and
$$\sum_{k=1}^{p-1}2^{k-1}C_k^{(2)}=(-1)^{(p-1)/2}-1 (mod p),$$ where
$C_k^{(2)}=binomial(3k,k)/(2k+1)$ is the $k$th Catalan number of order 2.

Let $p$ be a prime and let $a$ be a positive integer. In this paper we
investigate $\sum_{k=0}^{p^a-1}\binom[(h+1)k,k+d]/m^k$ modulo a prime $p$,
where $d$ and $m$ are integers with $-h<d<=p^a$ and $m\not=0 (mod p)$. We also
study congruences involving higher-order Catalan numbers
$C_k^{(h)}=\binom[(h+1)k,k]/(hk+1)$. Our tools include linear recurrences and
the theory of cubic residues. Here are some typical results in the paper.