multiplicative encoding

The multiplicative encoding of a finite sequence  $a$ of positive integers $k$ long is the product  of the first $k$ primes with the members of the sequence used as exponents, thus

 $\prod_{i=1}^{k}{p_{i}}^{a_{i}},$

with $p_{i}$ being the $i$th prime number  . For example, the fourth row of Pascal’s triangle is 1, 3, 3, 1. The multiplicative encoding is $2^{1}3^{3}5^{3}7^{1}=47250$.

Another use is in logic, such as Kurt Gödel encoding a logical proposition  as a single integer. As an example, Nagel and Newman convert $(\exists x)(x=sy)$ to the integer sequence 8, 4, 13, 9, 8, 13, 5, 7, 17, 9, and by multiplicative encoding to the single integer 172225505803959398742621651659678877886965404082311908389214945877004912002249920215937500000000.

References

• 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
Title multiplicative encoding MultiplicativeEncoding 2013-03-22 17:43:43 2013-03-22 17:43:43 PrimeFan (13766) PrimeFan (13766) 5 PrimeFan (13766) Definition msc 05C38