**Prime Power Representation of Integers**

There are lots of ways to represent numbers, base 10 and base 2 (binary) being examples. This one is a bit more sophisticated..

Every integer can be written as a product of primes. In fact, if p1,p2,p3,... is the prime number sequence (i.e. 2,3,5,7,..) then the general expression for an integer m is..

m=(p1^s1)*(p2^s2)*......*(pn^sn)

where s1,s2,s3,...,sn are integer powers.

Of course, if a prime pj is not involved in the prime decomposition we still include it, but we set sj=0 so pj^(sj)=1

So, any integer can be uniquely represented by its sequence or powers {sj}. Mathematicians call this representation the canonical representation. Canonical is just a fancy term for "standard".

An example will help..

567=(2)^0*(3)^4*(5)^0*(7)^1 so the prime power representation of 567 is the sequence {0,4,0,1}

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Content written and posted by Ken Abbott abbottsystems@gmail.com