2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/115285Rudin conjectured that there are never more than c N^(1/2) squares in an arithmetic progression of length N. Motivated by this surprisingly difficult problem we formulate more than twenty conjectures in harmonic analysis, analytic number theory, arithmetic geometry, discrete geometry and additive combinatorics (some old and some new) which each, if true, would shed light on Rudin's conjecture.21 pages, preliminary version. Comments welcomeNumber TheoryClassical Analysis and ODEs11N64text