  We prove that the 2-primary $\pi_{61}$ is zero. As a consequence, the
Kervaire invariant element $\theta_5$ is contained in the strictly defined
4-fold Toda bracket $\langle 2, \theta_4, \theta_4, 2\rangle$.
  Our result has a geometric corollary: the 61-sphere has a unique smooth
structure and it is the last odd dimensional case - the only ones are $S^1,
S^3, S^5$ and $S^{61}$.
  Our proof is a computation of homotopy groups of spheres. A major part of
this paper is to prove an Adams differential $d_3(D_3) = B_3$. We prove this
differential by introducing a new technique based on the algebraic and
geometric Kahn-Priddy theorems. The success of this technique suggests a
theoretical way to prove Adams differentials in the sphere spectrum inductively
by use of differentials in truncated projective spectra.
