All the solutions of the equation ∑i=1111xi=1 in distinct integers of the form xi∈3α5β7γxi∈3α5β7γ

It is shown: (i) the title equation has exactly 17 solutions all of which are exhibited, (ii) minx11=135minx11=135. When the values xixi are odd, but not all of the form 3α5β7γ3α5β7γ then: (iii) ∑i=1111xi=1 has at least fifteen solutions, five of which are demonstrated, (iv) x11=105x11=105 is attained, and the conjecture of J. Leech [R.K. Guy, Unsolved Problems in Number Theory, Springer, Berlin, NY, 1981] is confirmed.