In number theory, Moessner's theorem or Moessner's magic 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 in 1951; the first proof of its validity was given by Oskar Perron that same year. 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 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 ~.}
Example 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 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 ~.}
References
External links Polster, Burkard (17 July 2021). The Moessner Miracle. Why wasn't this discovered for over 2000 years?. Mathologer (short video documentary). Retrieved July 20, 2021 – via YouTube.