Numbers for which Euler Phi Function of 2n + 1 is less than that of 2n
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Theorem
The sequence of positive integers for which:
- $\map \phi {2 n + 1} < \map \phi {2 n}$
begins:
- $157, 262, 367, 412, \ldots$
This sequence is A001837 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
\(\ds 2 \times 157\) | \(=\) | \(\ds 314\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 157}\) | \(=\) | \(\ds 156\) | $\phi$ of $314$ | ||||||||||
\(\ds 2 \times 157 + 1\) | \(=\) | \(\ds 315\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 157 + 1}\) | \(=\) | \(\ds 144\) | $\phi$ of $315$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 157 + 1}\) | \(<\) | \(\ds \map \phi {2 \times 157}\) |
\(\ds 2 \times 262\) | \(=\) | \(\ds 524\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 262}\) | \(=\) | \(\ds 260\) | $\phi$ of $524$ | ||||||||||
\(\ds 2 \times 262 + 1\) | \(=\) | \(\ds 525\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 262 + 1}\) | \(=\) | \(\ds 240\) | $\phi$ of $525$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 262 + 1}\) | \(<\) | \(\ds \map \phi {2 \times 262}\) |
\(\ds 2 \times 367\) | \(=\) | \(\ds 734\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 367}\) | \(=\) | \(\ds 366\) | $\phi$ of $734$ | ||||||||||
\(\ds 2 \times 367 + 1\) | \(=\) | \(\ds 735\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 367 + 1}\) | \(=\) | \(\ds 336\) | $\phi$ of $735$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 367 + 1}\) | \(<\) | \(\ds \map \phi {2 \times 367}\) |
\(\ds 2 \times 412\) | \(=\) | \(\ds 824\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 412}\) | \(=\) | \(\ds 408\) | $\phi$ of $824$ | ||||||||||
\(\ds 2 \times 412 + 1\) | \(=\) | \(\ds 825\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 412 + 1}\) | \(=\) | \(\ds 400\) | $\phi$ of $825$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 412 + 1}\) | \(<\) | \(\ds \map \phi {2 \times 412}\) |
\(\ds 2 \times 472\) | \(=\) | \(\ds 944\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 472}\) | \(=\) | \(\ds 464\) | $\phi$ of $944$ | ||||||||||
\(\ds 2 \times 472 + 1\) | \(=\) | \(\ds 945\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 472 + 1}\) | \(=\) | \(\ds 432\) | $\phi$ of $945$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 472 + 1}\) | \(<\) | \(\ds \map \phi {2 \times 472}\) |
\(\ds 2 \times 487\) | \(=\) | \(\ds 974\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 487}\) | \(=\) | \(\ds 486\) | $\phi$ of $974$ | ||||||||||
\(\ds 2 \times 487 + 1\) | \(=\) | \(\ds 975\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 487 + 1}\) | \(=\) | \(\ds 480\) | $\phi$ of $975$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 487 + 1}\) | \(<\) | \(\ds \map \phi {2 \times 487}\) |
\(\ds 2 \times 577\) | \(=\) | \(\ds 1154\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 577}\) | \(=\) | \(\ds 576\) | $\phi$ of $1154$ | ||||||||||
\(\ds 2 \times 577 + 1\) | \(=\) | \(\ds 1155\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 577 + 1}\) | \(=\) | \(\ds 480\) | $\phi$ of $1155$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 577 + 1}\) | \(<\) | \(\ds \map \phi {2 \times 577}\) |
\(\ds 2 \times 682\) | \(=\) | \(\ds 1364\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 682}\) | \(=\) | \(\ds 600\) | $\phi$ of $1364$ | ||||||||||
\(\ds 2 \times 682 + 1\) | \(=\) | \(\ds 1365\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 682 + 1}\) | \(=\) | \(\ds 576\) | $\phi$ of $1365$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {2 \times 682 + 1}\) | \(<\) | \(\ds \map \phi {2 \times 682}\) |
This theorem requires a proof. In particular: that these are the smallest such. The philosophical significance of this result is not immediately clear. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $157$