Writing a mathematics paper
To add to @Buffy’s answer, in all math graduate programs I’m aware of, at least in the US, letters of recommendation are a requirement. So a “lack of letters of recommendation” is actually a guarantee that you will not get admitted. This is not something you can solve by publishing either one “simple” paper, or even many non-simple ones, except in the indirect sense that if you publish many papers then good letters of recommendation are likely to follow more or less automatically.

Keep in mind that this applies to all programs, not just “top-tier” ones.