Search for a tool
Möbius Function

Tool to calculate the value of the function μ (Mu) of Möbius (or Moebius) which has a value of -1, 0 or 1 according to its prime numbers decomposition.

Results

Möbius Function -

Tag(s) : Arithmetics

Share
Share
dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!


Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!


Feedback and suggestions are welcome so that dCode offers the best 'Möbius Function' tool for free! Thank you!

Möbius Function

Mobius μ(N) Calculator


Answers to Questions (FAQ)

What is the Mobius Mu function? (Definition)

The function $ μ(n) $, called the Möbius function (or Moebius), is defined for any integer $ n > 0 $ of the set $ \mathbb{N}* $ in the set of 3 values $ \{ -1, 0, 1 \} $.

$ μ(n) $ is $ 0 $ if $ n $ has for divisor a perfect square (other than 1)

$ μ(n) $ is $ 1 $ if $ n $ has for divisors an even number of prime numbers

$ μ(n) $ is $ -1 $ if $ n $ has for divisors an odd number of prime numbers

How to calculate the value of the Mobius function?

Automatic method: indicate the value $ n $ for which to calculate $ μ(n) $ in dCode (above)

Manual method: the image of $ μ(n) $ depends on the prime number decomposition of $ n $. If a prime number appears several times in the decomposition, then $ μ(n) = 0 $, otherwise, if the decomposition has an even number of prime numbers, then $ μ(n) = 1 $ and otherwise with an odd number of prime numbers $ μ(n) = -1 $.

Example: $ 12 = 2 \times 2 \times 3 $ so $ μ(12) = 0 $ because $ 2 $ appears twice, and so $ 12 $ is divisible by $ 4 $, a perfect square

Example: $ 1234 = 2 \times 617 $ therefore $ μ(12) = 1 $ because the decomposition has 2 distinct primes (2 is an even number)

Example: $ 12345 = 3 \times 5 \times 823 $ so $ μ(12) = -1 $ because the decomposition has 3 distinct prime numbers (3 is an odd number)

Source code

dCode retains ownership of the "Möbius Function" source code. Except explicit open source licence (indicated Creative Commons / free), the "Möbius Function" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Möbius Function" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Möbius Function" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.

Cite dCode

The copy-paste of the page "Möbius Function" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Möbius Function on dCode.fr [online website], retrieved on 2024-07-18, https://www.dcode.fr/mobius-function

Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Questions / Comments

Feedback and suggestions are welcome so that dCode offers the best 'Möbius Function' tool for free! Thank you!


https://www.dcode.fr/mobius-function
© 2024 dCode — El 'kit de herramientas' definitivo para resolver todos los juegos/acertijos/geocaching/CTF.
 
Feedback