Numbers which are Sum of Increasing Powers of Digits

From ProofWiki
Jump to navigation Jump to search

Theorem

The following integers are the sum of the increasing powers of their digits taken in order begin as follows:

$0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 89, 135, 175, 518, 598, 1306, 1676, 2427, 2 \, 646 \, 798, 12 \, 157 \, 692 \, 622 \, 039 \, 623 \, 539$

This sequence is A032799 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Single digit integers are trivial:

$d^1 = d$

for all $d \in \Z$.


Then we have:

\(\ds 8^1 + 9^2\) \(=\) \(\ds 8 + 81\) \(\ds = 89\)


\(\ds 1^1 + 3^2 + 5^3\) \(=\) \(\ds 1 + 9 + 125\) \(\ds = 135\)
\(\ds 1^1 + 7^2 + 5^3\) \(=\) \(\ds 1 + 49 + 125\) \(\ds = 175\)
\(\ds 5^1 + 1^2 + 8^3\) \(=\) \(\ds 5 + 1 + 512\) \(\ds = 518\)
\(\ds 5^1 + 9^2 + 8^3\) \(=\) \(\ds 5 + 81 + 512\) \(\ds = 598\)


\(\ds 1^1 + 3^2 + 0^3 + 6^4\) \(=\) \(\ds 1 + 9 + 0 + 1296\) \(\ds = 1306\)
\(\ds 1^1 + 6^2 + 7^3 + 6^4\) \(=\) \(\ds 1 + 36 + 343 + 1296\) \(\ds = 1676\)
\(\ds 2^1 + 4^2 + 2^3 + 7^4\) \(=\) \(\ds 2 + 16 + 8 + 2401\) \(\ds = 2427\)


Then:

\(\ds \) \(\) \(\ds 2^1 + 6^2 + 4^3 + 6^4 + 7^5 + 9^6 + 8^7\)
\(\ds \) \(=\) \(\ds 2 + 36 + 64 + 1296 + 16 \, 807 + 531 \, 441 + 1 \, 097 \, 152\)
\(\ds \) \(=\) \(\ds 2 \, 646 \, 798\)


and finally:

\(\ds \) \(\) \(\ds 1 + 2^2 + 1^3 + 5^4 + 7^5 + 6^6 + 9^7 + 2^8 + 6^9 + 2^{10} + 2^{11} + 0^{12} + 3^{13} + 9^{14} + 6^{15} + 2^{16} + 3^{17} + 5^{18} + 3^{19} + 9^{20}\)
\(\ds \) \(=\) \(\ds 1 + 4 + 1 + 625 + 16 \, 807 + 46 \, 656 + 4 \, 782 \, 969 + 256 + 10 \, 077 \, 696 + 1024 + 2048 + 0 + 1 \, 594 \, 323\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds 22 \, 876 \, 792 \, 454 \, 961 + 470 \, 184 \, 984 \, 576 + 65 \, 536 + 129 \, 140 \, 163 + 3 \, 814 \, 697 \, 265 \, 625\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds 1 \, 162 \, 261 \, 467 + 12 \, 157 \, 665 \, 459 \, 056 \, 928 \, 801\)
\(\ds \) \(=\) \(\ds 12 \, 157 \, 692 \, 622 \, 039 \, 623 \, 539\)


This theorem requires a proof.
In particular: It remains to be proved that there are no more like this.
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

Retrieved from "https://proofwiki.org/w/index.php?title=Numbers_which_are_Sum_of_Increasing_Powers_of_Digits&oldid=408740"