Graham's number stable digits: An exact solution
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908533854830592 |
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| author | Ripà, Marco |
| author_facet | Ripà, Marco |
| contents | In the decimal numeral system, we prove that the well-known Graham's number, $G := \! ^{n}3$ (i.e., $3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}$ ($n$ times)), and any base $3$ tetration whose hyperexponent is larger than $n$ share the same $\operatorname{slog}_3(G) - 1$ rightmost digits (where $\operatorname{slog}$ indicates the integer super-logarithm). This is an exact result since the $\operatorname{slog}_3(G)$-th rightmost digit of $G$ differs from the $\operatorname{slog}_3(G)$-th rightmost digit of $^{n+1}3$. Furthermore, we show that the $\operatorname{slog}_3(^{n}3)$-th least significant digit of the difference between Graham's number and any base $3$ tetration whose integer hyperexponent exceeds $n$ is $4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_00015 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Graham's number stable digits: An exact solution Ripà, Marco General Mathematics 11A07, 11F33 In the decimal numeral system, we prove that the well-known Graham's number, $G := \! ^{n}3$ (i.e., $3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}$ ($n$ times)), and any base $3$ tetration whose hyperexponent is larger than $n$ share the same $\operatorname{slog}_3(G) - 1$ rightmost digits (where $\operatorname{slog}$ indicates the integer super-logarithm). This is an exact result since the $\operatorname{slog}_3(G)$-th rightmost digit of $G$ differs from the $\operatorname{slog}_3(G)$-th rightmost digit of $^{n+1}3$. Furthermore, we show that the $\operatorname{slog}_3(^{n}3)$-th least significant digit of the difference between Graham's number and any base $3$ tetration whose integer hyperexponent exceeds $n$ is $4$. |
| title | Graham's number stable digits: An exact solution |
| topic | General Mathematics 11A07, 11F33 |
| url | https://arxiv.org/abs/2411.00015 |