When your friend is not that smart at Maths.

Assume picking such 125 numbers is possible. Notice that each of the chosen numbers is in one of 125 intervals : $\langle 1;2^{1025});\langle 2^{1025};3^{1025}) \cdots\langle 123^{1025};124^{1025}), \{124^{1025}\}$ , and from the pigeonhole principle some two chosen numbers a>b are in one interval $\langle x^{1025};(x+1)^{1025})$ for some integer x. Then $|\sqrt[1025]a-\sqrt[1025]b| = \sqrt[1025]a-\sqrt[1025]b < \sqrt[1025]{(x+1)^{1025}}-\sqrt[1025]{x^{1025}}=1$ . That is a contradiction and thus 125 numbers with conditions from the exercice can't be picked.