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In this article, we show that $x+y=z$ has only finitely many solutions $(x,y,z)\\in \\mathrm{PS}(\\alpha)^3$ for almost every $\\alpha>3$. Furthermore, we show that $\\mathrm{PS}(\\alpha)$ has only finitely many arithmetic progressions of length $3$ for almost every $\\alpha>10$. 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Let $\\mathrm{PS}(\\alpha)$ denote the set of all those terms. In this article, we show that $x+y=z$ has only finitely many solutions $(x,y,z)\\in \\mathrm{PS}(\\alpha)^3$ for almost every $\\alpha>3$. Furthermore, we show that $\\mathrm{PS}(\\alpha)$ has only finitely many arithmetic progressions of length $3$ for almost every $\\alpha>10$. 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