  Bondy and Vince showed that every graph with minimum degree at least three
contains two cycles of lengths differing by one or two.We prove the following
average degree counterpart that every $n$-vertex graph $G$ with at least
$\frac52(n-1)$ edges, unless $4|(n-1)$ and every block of $G$ is a clique
$K_5$, contains two cycles of consecutive even lengths. Our proof is mainly
based on structural analysis, and a crucial step which may be of independent
interest shows that the same conclusion holds for every 3-connected graph with
at least 6 vertices. This solves a special case of a conjecture of
Verstra\"ete. The quantitative bound is tight and also provides the optimal
extremal number for cycles of length two modulo four.
