PRIME , COMPOSITE, AND SUPER-COMPOSITE NUMBERS Any integer N greater than one is either a prime or composite number. Primes have no divisors except one and N itself while composites will have multiple divisors. It is always possible to represent a composite number as the product of primes taken to specified powers. That is- n ak N ( pk ) where p1 2, p2 3, p3 5, p4 7, p5 11, …. k1 Several years ago while examining the sum of all divisors of a number N we came up with a new function defined as- (N ) N 1 f (N ) N where (N) is the sigma function of number theory. This function f(N), which applies only to integer N, has the interesting property that whenever N is a prime then f(N)=0 and when it is a composite f(N) will be greater than zero. We have termed this point function the number fraction. Our purpose here is to use the properties of the number fraction as a means to quickly distinguish between primes and composites and also to determine the conditions under which a composite becomes a super-composite. We start by presenting a graph of f(N) versus N in the limited range 3<N<61. It looks like this- We see there the numbers for which f(N)=0. These represent the primes p=3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59, and 61. Also we have the composite numbers when f(N)>0 with some F(N)s reaching local maxima. These maxima at N=12, 24, 36, 48,and 60 will be referred to as super-composites. Their main characteristic is that they have a large number of factors compared to their immediate neighbors. As N increases in value the super-composites become ever more apparent as shown in the following graph of f(N) in the vicinity of N=55440- We see there a peak value of f(N) equal to 3.1869949 and two neighboring primes with f(N1)=0. When two primes are separated from each other by 2 they are often referred to as twin primes. We next look at some further results which may be generated by f(N). Consider a prime number p and write down the following- f ( p) 0 1 f ( p 2 ) p 1 p f ( p3 ) p 2 1 p p 2 f ( p 4 ) p3 On generalizing this we have- n2 k p n1 ( p 1) f ( p n ) k0 p n1 ( p 1) p n1 Combining of f(p2) and f(p3) we get- [ f ( p 2 ) 1] 1 f ( p3 ) This is an interesting result. It shows for any prime number five or greater the last quotient must be equal to one. It suggests a new Prime Number Function- [ f (N 2 ) 1] [ (N 2 ) 1] F(N ) f (N 3 ) [ (N 3 ) N 3 1] Let us demonstrate the ability to distinguish between prime and compound numbers using this last formula. Consider N=91 and N=127. The function evaluates to- F(91)=5215/992140.05256 and F(127)=1 Thus 91 is a compound number 91=7x13 while 127 is a prime number. One suspects that compound numbers will always produce values of F(N) less than one. To support this conjecture we present the value of F(N) in graphical form over the range 2 through 40. The graph is generated by the one line computer program- listplot([seq([x,F],x=2..40)]); and yields- The graph clearly shows all prime numbers have a function value F(N)=1. For compound numbers it appears to always have values less than one. Note the presence of twin primes at [5,7], [11,13], [29,31] . As already found in an earlier note, the twin primes are given by [5+6n, 7+6n] for specified values of the integer n. The smallest F(N) in the range 20<N<40 occurs at F(36)=625/15359=0.04069275343. Thirty-six is one of the super- composites given above. It is actually easier to find using f(N) instead of F(N). To distinguish super-composites from composites one typically looks at graphs of f(N) versus N over a restricted finite range of N and then identifies super-composites by their local maximum. A few of these super-primes written out in their prime product form are- 36 22 32 180 22 32 5 151120 24 33 57 55440 24 32 5711 A common characteristic of these super-composites is that they contain only the lowest primes taken at high but decreasing powers ak as pk increases. It suggests, for example, that 26 34 53 648000 should be a super-composite. Checking this out in a band of 10 about 648000 we get the following graph- So N=648000 is indeed a super-composite with its f (N)=2.69946 value towering over its neighbors. The number 648007 is a prime since f(648007)=0. However, N=648001 has finite but small value f(648001)=0.006941 and so is a composite given by 149x4349. Such near zero number fractions usually indicate is a semi-prime where N=pq , with p and q primes. Large Semi-primes become of interest in connection with public key cryptography. A final question which arises deals with the magnitude of f(N) for a large N super- primes. Does the value of F(N) remain finite for such numbers? One can argue that- (N ) N 1 (N ) f (N ) for N N N But the ratio (N)/N is found for all integers considered up to 100 digit length to have a finite value not much higher than 7 . Thus the maximum f(N) corresponding to a super- composite should also have a finite f(N )value. August 28, 2016 .