Sequence of Syracuse : Extensions of types 3n+b and 5n+1
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| Format: | Recurso digital |
| Sprache: | Englisch |
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2025
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| _version_ | 1866902148861657088 |
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| author | BALLASI, Jacques |
| author_facet | BALLASI, Jacques |
| contents | <p>We investigate two families of extensions of the well-known Syracuse sequence. In the standard case, starting from an initial integer $v_0 > 0$, the sequence $(v_n)$ is defined by the recurrence: $v_{n+1} = (3v_n + 1)/2$ if $v_n$ is odd, and $v_n/2$ if $v_n$ is even.</p> <p>For the generalizations studied here, we denote the sequence by $(V_n)$ to clearly distinguish it from the classical case. In all cases, the rule applied to even integers remains $V_{n+1} = V_n/2$. We consider two types of extensions:</p> <p>- $3n + b$ extensions: for a fixed odd integer $b$, the transformation becomes $V_{n+1} = (3V_n + b)/2$ when $V_n$ is odd;<br>- The $5n + 1$ extension: the rule becomes $V_{n+1} = (5V_n + 1)/2$ when $V_n$ is odd.</p> <p>Building upon results established in the classical case (when $b = 1$), we show that:<br>- for every odd integer $b$, the $3n + b$ extension admits only finitely many cycles, all of which can be detected by testing initial values $V_0 \leq 2^{f(b)} \cdot |b|$. Moreover, no sequence diverges. In particular, when $|b| < 2^{10}$, the bound $f(b) = 48$ suffices;<br> - in contrast, the $5n + 1$ extension admits at least one divergent orbit.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_16945277 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Sequence of Syracuse : Extensions of types 3n+b and 5n+1 BALLASI, Jacques Collatz conjecture 3x+1 problem <p>We investigate two families of extensions of the well-known Syracuse sequence. In the standard case, starting from an initial integer $v_0 > 0$, the sequence $(v_n)$ is defined by the recurrence: $v_{n+1} = (3v_n + 1)/2$ if $v_n$ is odd, and $v_n/2$ if $v_n$ is even.</p> <p>For the generalizations studied here, we denote the sequence by $(V_n)$ to clearly distinguish it from the classical case. In all cases, the rule applied to even integers remains $V_{n+1} = V_n/2$. We consider two types of extensions:</p> <p>- $3n + b$ extensions: for a fixed odd integer $b$, the transformation becomes $V_{n+1} = (3V_n + b)/2$ when $V_n$ is odd;<br>- The $5n + 1$ extension: the rule becomes $V_{n+1} = (5V_n + 1)/2$ when $V_n$ is odd.</p> <p>Building upon results established in the classical case (when $b = 1$), we show that:<br>- for every odd integer $b$, the $3n + b$ extension admits only finitely many cycles, all of which can be detected by testing initial values $V_0 \leq 2^{f(b)} \cdot |b|$. Moreover, no sequence diverges. In particular, when $|b| < 2^{10}$, the bound $f(b) = 48$ suffices;<br> - in contrast, the $5n + 1$ extension admits at least one divergent orbit.</p> |
| title | Sequence of Syracuse : Extensions of types 3n+b and 5n+1 |
| topic | Collatz conjecture 3x+1 problem |
| url | https://doi.org/10.5281/zenodo.16945277 |