We describe several methods which permit one to search for big integral points on certain elliptic curves, i.e., for integral solutions (x, y) of certain Diophantine equations of the form
y
2
=
x
3
+
a
x
+
b
(
a
,
b
∈
Z
)
{y^2} = {x^3} + ax + b\;(a,b \in {\mathbf {Z}})
in a large range
|
x
|
,
|
y
|
⩽
B
|x|,|y| \leqslant B
, in time polynomial in
log
log
B
\log \log B
. We also give a number of individual examples and of parametric families of examples of specific elliptic curves having a relatively large integral point.