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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
1
n
,
2
n
,
3
n
,
4
n
,
⋯
,
{\displaystyle 1^{n},2^{n},3^{n},4^{n},\cdots ~,}
with
n
≥
1
,
{\displaystyle n\geq 1~,}
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
n
=
2
{\displaystyle n=2}
, one can remove every even number, resulting in
(
1
,
3
,
5
,
7
⋯
)
{\displaystyle (1,3,5,7\cdots )}
, and then add each odd number to the sum of all previous elements, providing
(
1
,
4
,
9
,
16
,
⋯
)
=
(
1
2
,
2
2
,
3
2
,
4
2
⋯
)
{\displaystyle (1,4,9,16,\cdots )=(1^{2},2^{2},3^{2},4^{2}\cdots )}
.
Construction [ edit ]
Write down every positive integer and remove every
n
{\displaystyle n}
-th element, with
n
{\displaystyle n}
a positive integer. Build a new sequence of partial sums with the remaining numbers. Continue by removing every
(
n
−
1
)
{\displaystyle (n-1)}
-st element in the new sequence and producing a new sequence of partial sums. For the sequence
k
{\displaystyle k}
, remove the
(
n
−
k
+
1
)
{\displaystyle (n-k+1)}
-st elements and produce a new sequence of partial sums.
The procedure stops at the
n
{\displaystyle n}
-th sequence. The remaining sequence will correspond to
1
n
,
2
n
,
3
n
,
4
n
⋯
.
{\displaystyle 1^{n},2^{n},3^{n},4^{n}\cdots ~.}
[4] [5]
Example [ edit ]
The initial sequence is the sequence of positive integers,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
,
11
,
12
,
13
,
14
,
15
,
16
⋯
.
{\displaystyle 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16\cdots ~.}
For
n
=
4
{\displaystyle n=4}
, we remove every fourth number from the sequence of integers and add up each element to the sum of the previous elements
1
,
2
,
3
,
5
,
6
,
7
,
9
,
10
,
11
,
13
,
14
,
15
⋯
→
1
,
3
,
6
,
11
,
17
,
24
,
33
,
43
,
54
,
67
,
81
,
96
⋯
{\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 }
Now we remove every third element and continue to add up the partial sums
1
,
3
,
11
,
17
,
33
,
43
,
67
,
81
⋯
→
1
,
4
,
15
,
32
,
65
,
108
,
175
,
256
⋯
{\displaystyle 1,3,11,17,33,43,67,81\cdots \to 1,4,15,32,65,108,175,256\cdots }
Remove every second element and continue to add up the partial sums
1
,
15
,
65
,
175
⋯
→
1
,
16
,
81
,
256
⋯
{\displaystyle 1,15,65,175\cdots \to 1,16,81,256\cdots }
,
which recovers
1
4
,
2
4
,
3
4
,
4
4
,
⋯
{\displaystyle 1^{4},2^{4},3^{4},4^{4},\cdots }
.
Variants [ edit ]
If the triangular numbers are removed instead, a similar procedure leads to the sequence of factorials
1
!
,
2
!
,
3
!
,
4
!
,
⋯
.
{\displaystyle 1!,2!,3!,4!,\cdots ~.}
[1]
References [ edit ]
^
Moessner, Alfred (1951). "Eine Bemerkung über die Potenzen der natürlichen Zahlen" [A note on the powers of the natural numbers]. Sitzungsberichte (in German). 3 .
^
Oskar, Perron (1951). "Beweis des Moessnerschen Satzes" [Proof of Moessner's theorem]. Sitzungsberichte (in German). 4 .
^ a b
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 . hdl :2066/111198 . S2CID 8799795 .
^ Weisstein, Eric W. "Moessner's Theorem" . mathworld.wolfram.com . Retrieved 2021-07-20 .
External links [ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Moessner%27s_theorem&oldid=1181580818 "
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