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multiplicative encoding
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(Definition)
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The multiplicative encoding of a finite sequence  of positive integers  long is the product of the first  primes with the members of the sequence used as exponents, thus
with  being the  th prime number. For example, the fourth row of Pascal's triangle is 1, 3, 3, 1. The multiplicative encoding is
 .
Encryption is not the purpose of multiplicative encoding, as the original sequence is easily retrieved even with trial division. However, there are applications in combinatorics. Neil Sloane, for example, encodes the most famous number triangles as multiplicative encodings of the rows in order. While the resulting sequence for Pascal's triangle does not consist of squarefree numbers (save the first two), it does contain only singly even numbers.
Another use is in logic, such as Kurt Gödel encoding a logical proposition as a single integer. As an example, Nagel and Newman convert
 to the integer sequence 8, 4, 13, 9, 8, 13, 5, 7, 17, 9, and by multiplicative encoding to the single integer 172225505803959398742621651659678877886965404082311908389214945877004912002249920215937500000000.
- 1
- Ernest Nagel & James Newman, Gödel's Proof. New York: New York University Press (2001): 75 - 76
- 2
- Neil Sloane, The Encyclopedia of Integer Sequences. New York: Academic Press (1995): M1722
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"multiplicative encoding" is owned by PrimeFan.
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Cross-references: proposition, logic, singly even numbers, contain, squarefree, order, triangles, number, Neil Sloane, applications, trial division, Pascal's triangle, row, exponents, sequence, primes, product, integers, positive, finite sequence
There is 1 reference to this entry.
This is version 2 of multiplicative encoding, born on 2008-01-05, modified 2008-01-16.
Object id is 10176, canonical name is MultiplicativeEncoding.
Accessed 420 times total.
Classification:
| AMS MSC: | 05C38 (Combinatorics :: Graph theory :: Paths and cycles) |
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Pending Errata and Addenda
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