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JavaScript (Node.js), 87 bytes f=(a,b=2,c=a-b,d=(a,b=2)=>b<a?a%b&&d(a,b+1):1,e=a=>d(++a)?a:e(a))=>d(c)?[b,c]:f(a,e(b)) Try it online!
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# [JavaScript (Node.js)], 87 bytes <!-- language-all: lang-javascript --> f=(a,b=2,c=a-b,d=(a,b=2)=>b<a?a%b&&d(a,b+1):1,e=a=>d(++a)?a:e(a))=>d(c)?[b,c]:f(a,e(b)) [Try it online!][TIO-kkyoq89c] [JavaScript (Node.js)]: https://nodejs.org [TIO-kkyoq89c]: https://tio.run/##LUy7DoIwFN35ii5Cb1oIEOMAXpj8CsJw@0AhhBowLsZvr4W4nHfORG/a9Do@X@nijPV@QE5SYSk1Uqqk@VvARl2ppZOKY7NHooCqkBYJG8OFIGipspwAdq@h7ZTUfTWEqeUKwNddxNhZBrjsUB6yKPOD8jxwnw1uvZF@cGLYsE8otFs2N9tsdnceriCb3LjwhCUAdfSF2v8A "JavaScript (Node.js) – Try It Online"