Moessner's theorem
In number theory, Moessner's theorem or Moessner's magic[1] is related to an arithmetical algorithm to produce an infinite sequence of the exponents of positive integers [math]\displaystyle{ 1^n, 2^n, 3^n, 4^n, \cdots ~, }[/math] with [math]\displaystyle{ n \geq 1 ~, }[/math] by recursively manipulating the sequence of integers algebraically. The algorithm was first published by Alfred Moessner[2] in 1951; the first proof of its validity was given by Oskar Perron[3] that same year.[4]
For example, for [math]\displaystyle{ n=2 }[/math], one can remove every even number, resulting in [math]\displaystyle{ (1,3,5,7\cdots) }[/math], and then add each odd number to the sum of all previous elements, providing [math]\displaystyle{ (1,4,9,16,\cdots)=(1^2,2^2,3^2,4^2\cdots) }[/math].
Construction
Write down every positive integer and remove every [math]\displaystyle{ n }[/math]-th element, with [math]\displaystyle{ n }[/math] a positive integer. Build a new sequence of partial sums with the remaining numbers. Continue by removing every [math]\displaystyle{ (n-1) }[/math]-st element in the new sequence and producing a new sequence of partial sums. For the sequence [math]\displaystyle{ k }[/math], remove the [math]\displaystyle{ (n-k+1) }[/math]-st elements and produce a new sequence of partial sums.
The procedure stops at the [math]\displaystyle{ n }[/math]-th sequence. The remaining sequence will correspond to [math]\displaystyle{ 1^n, 2^n, 3^n, 4^n \cdots~. }[/math][4][5]
Example
The initial sequence is the sequence of positive integers,
- [math]\displaystyle{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 \cdots ~. }[/math]
For [math]\displaystyle{ n=4 }[/math], we remove every fourth number from the sequence of integers and add up each element to the sum of the previous elements
- [math]\displaystyle{ 1,2,3,5,6,7,9,10,11,13,14,15 \cdots \to 1,3,6,11,17,24,33,43,54,67,81,96 \cdots }[/math]
Now we remove every third element and continue to add up the partial sums
- [math]\displaystyle{ 1,3,11,17,33,43,67,81 \cdots \to 1,4,15,32,65,108,175,256 \cdots }[/math]
Remove every second element and continue to add up the partial sums
- [math]\displaystyle{ 1,15,65,175 \cdots \to 1,16,81,256 \cdots }[/math],
which recovers [math]\displaystyle{ 1^4, 2^4,3^4,4^4, \cdots }[/math].
Variants
If the triangular numbers are removed instead, a similar procedure leads to the sequence of factorials [math]\displaystyle{ 1!, 2!,3!,4!,\cdots~. }[/math][1]
References
- ↑ 1.0 1.1 Conway, John H.; Guy, Richard (2012-12-06). The Book of Numbers. Springer Science & Business Media. ISBN 978-1-4612-4072-3. https://books.google.com/books?id=rfLSBwAAQBAJ&q=moessner+magic&pg=PA63.
- ↑ Moessner, Alfred (1951). "Eine Bemerkung über die Potenzen der natürlichen Zahlen". Sitzungsberichte 3. https://publikationen.badw.de/de/003383662.
- ↑ "Beweis des Moessnerschen Satzes". Sitzungsberichte 4. 1951. https://publikationen.badw.de/de/043542577.
- ↑ 4.0 4.1 Kozen, Dexter; Silva, Alexandra (2013). "On Moessner's Theorem". The American Mathematical Monthly 120 (2): 131. doi:10.4169/amer.math.monthly.120.02.131. https://www.tandfonline.com/doi/full/10.4169/amer.math.monthly.120.02.131.
- ↑ Weisstein, Eric W.. "Moessner's Theorem". https://mathworld.wolfram.com/MoessnersTheorem.html.
External links
- The Moessner Miracle. Why wasn't this discovered for over 2000 years?. Mathologer (short video documentary). 17 July 2021. Retrieved July 20, 2021 – via YouTube.
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