Editing Openai/68ec50da-cf00-8005-b5f6-b683506e5853 (section)

===== If r∉{±7}r\not\in\{\pm7\}r∈{±7}, then for a,b∈Ara,b\in A_ra,b∈Ar we have ab+1≡r2+1≢0(mod25)ab+1\equiv r^2+1\not\equiv0\pmod{25}ab+1≡r2+1≡0(mod25). Applying Lemma 1 with t=rt=rt=r shows that for each fixed a∈Ara\in A_ra∈Ar a positive density (≥δ−o(1)\ge\delta-o(1)≥δ−o(1)) of the b∈Arb\in A_rb∈Ar make ab+1ab+1ab+1 squarefree, contradicting (P) unless mr=o(N)m_r=o(N)mr=o(N). =====

Therefore, the only 252525-classes that can contain ≫N\gg N≫N elements of AAA are the two fixed points of the involution x↦−x−1x\mapsto -x^{-1}x↦−x−1 modulo 252525, namely r=7r=7r=7 and r=18r=18r=18, because precisely for those two classes we have r2≡−1(mod25)r^2\equiv-1\pmod{25}r2≡−1(mod25) and thus 25∣ab+125\mid ab+125∣ab+1 for every a,b≡ra,b\equiv ra,b≡r.