| Summary: | In this note, we show that, for any real number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>∈</mo><mo>[</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, any finite set of positive integers <i>K</i> and any integer <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>s</mi><mn>1</mn></msub><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>, the sequence of integers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><msub><mi>s</mi><mn>2</mn></msub><mo>,</mo><msub><mi>s</mi><mn>3</mn></msub><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula> satisfying <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>s</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>s</mi><mi>i</mi></msub><mo>∈</mo><mi>K</mi></mrow></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>s</mi><mi>i</mi></msub></semantics></math></inline-formula> is a prime number, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>≤</mo><msub><mi>s</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>≤</mo><mi>τ</mi><msub><mi>s</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>s</mi><mi>i</mi></msub></semantics></math></inline-formula> is a composite number, is bounded from above. The bound is given in terms of an explicit constant depending on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>,</mo><msub><mi>s</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> and the maximal element of <i>K</i> only. In particular, if <i>K</i> is a singleton set and for each composite <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>s</mi><mi>i</mi></msub></semantics></math></inline-formula> the integer <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>s</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> in the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>2</mn><mo>,</mo><mi>τ</mi><msub><mi>s</mi><mi>i</mi></msub><mo>]</mo></mrow></semantics></math></inline-formula> is chosen by some prescribed rule, e.g., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>s</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> is the largest prime divisor of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>s</mi><mi>i</mi></msub></semantics></math></inline-formula>, then the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><msub><mi>s</mi><mn>2</mn></msub><mo>,</mo><msub><mi>s</mi><mn>3</mn></msub><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula> is periodic. In general, we show that the sequences satisfying the above conditions are all periodic if and only if either <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>=</mo><mo>{</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>∈</mo><mo>[</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mo>)</mo></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>=</mo><mo>{</mo><mn>2</mn><mo>}</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>∈</mo><mo>[</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mfrac><mn>5</mn><mn>9</mn></mfrac><mo>)</mo></mrow></semantics></math></inline-formula>.
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