Moessner's theorem explained

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,} … ~, with 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 , one can remove every even number, resulting in ( 1 , 3 , 5 , 7 … ) , 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 … )} .

Construction

Write down every positive integer and remove every

n -th element, with n a positive integer. Build a new sequence of partial sums with the remaining numbers. Continue by removing every (n -1 ) -st element in the new sequence and producing a new sequence of partial sums. For the sequence k , remove the (n - k +1 ) -st elements and produce a new sequence of partial sums.

The procedure stops at the

n -th sequence. The remaining sequence will correspond to 1 ^{n,} 2 ^{n,} 3 ^{n,} 4 ^{n} … ~. ^{[4] [5]}

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 … ~.

For

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 … \to 1 , 3 , 6 , 11 , 17 , 24 , 33 , 43 , 54 , 67 , 81 , 96 …

Now we remove every third element and continue to add up the partial sums

1 , 3 , 11 , 17 , 33 , 43 , 67 , 81 … \to 1 , 4 , 15 , 32 , 65 , 108 , 175 , 256 …

Remove every second element and continue to add up the partial sums

1 , 15 , 65 , 175 … \to 1 , 16 , 81 , 256 … ,which recovers 1 ^{4,} 2 ^{4,3} ^{4,4} ^{4,} … .

Variants

1 !, 2 !, 3 !, 4 !, … ~. ^[1]

External links

• Polster . Burkard . Burkard Polster . 17 July 2021 . The Moessner Miracle. Why wasn't this discovered for over 2000 years? . short video documentary . Mathologer . YouTube . July 20, 2021.

Notes and References

1. Book: Conway . John H. . Guy . Richard . 2012-12-06 . The Book of Numbers . Springer Science & Business Media . 978-1-4612-4072-3 . en .
2. Moessner . Alfred . 1951 . Eine Bemerkung über die Potenzen der natürlichen Zahlen . de . A note on the powers of the natural numbers . Sitzungsberichte . 3 .
3. Oskar . Perron . Oskar Perron . 1951 . Beweis des Moessnerschen Satzes . de . Proof of Moessner's theorem . Sitzungsberichte . 4 .
4. Kozen . Dexter . Silva . Alexandra . 2013 . On Moessner's Theorem . . 120 . 2 . 131 . 8799795 . 10.4169/amer.math.monthly.120.02.131 . 2066/111198 . free .
5. Web site: Weisstein . Eric W. . Moessner's Theorem . mathworld.wolfram.com . en . 2021-07-20.