T-sequence apparatus and method for general deterministic polynomial-time primality testing and composite factoring

Abstract

Using a new mathematical technique called the T-sequence, the inventor has discovered a powerful primality testing method that meets all four conditions above. A similar approach can be applied to perform fast factoring for numerous special cases, a method that can, in all liklihood, be extended to the general case, making possible a general and fast factoring algorithm. (Researchers heretofore have been able to factor only in sub-exponential time, never in polynomial time.) The same T-sequence can be used to construct a prime number formula (long sought after but never achieved) and a good random number generator. The former can be used to generate infinitely many prime numbers of any size efficiently, and the latter can generate non-periodic and absolutely chaotic random numbers. These aft numbers are widely used in all areas of industrial and scientific simulations. In general, the T-sequence can be used to handle efficiently the fundamental problems concerning prime numbers (which include primality testing, factoring, prime number formula, infinite-pattern prime problem, etc.).

what is claimed is:  
     
         1 . A computer-implemented method, comprising: 
 determining at least one element of a non-montonic sequence, the non-montonic sequence being one of a family of related non-montonic sequences;    using at least said element, determining at least one property of a number; and    depending on said property, taking an action the effect of which is to enhance or degrade data security within a computer system or network.    
     
     
         2 . The method of  claim 1 , wherein said property is primality.  
     
     
         3 . The method of  claim 1 , wherein said number is a composite number, and said property is a factor of said number.  
     
     
         4 . The method of  claim 1 , wherein said family of related non-montonic sequences is defined as follows:  
       
         
           
             
               
                 
                   T 
                   0 
                   l 
                 
                 = 
                 2 
               
               , 
               
                 
                   T 
                   1 
                   l 
                 
                 = 
                 
                   
                     l 
                      
                     
                         
                     
                      
                     and 
                      
                     
                         
                     
                      
                     
                       T 
                       
                         n 
                         + 
                         1 
                       
                       l 
                     
                   
                   = 
                   
                     
                       l 
                       · 
                       
                         T 
                         n 
                         l 
                       
                     
                     - 
                     
                       T 
                       
                         n 
                         - 
                         1 
                       
                       l 
                     
                   
                 
               
               , 
             
           
           
           
               
           
         
         where the subscript denotes the nth term while the superscript denotes the order l.  
       
     
     
         5 . A prime number generator, comprising: 
 means for generating candidate numbers by forming at least one of sums and differences of a given number and a series of prime numbers; and    means for deterministically evaluating primality of each of the candidate numbers in polynomial time.    
     
     
         6 . The apparatus of  claim 5 , wherein said means for deterministically evaluating primality comprises means for determining at least one element of a non-montonic sequence, the non-montonic sequence being one of a family of related non-montonic sequences.  
     
     
         7 . The apparatus of  claim 6 , wherein said family of related non-montonic sequences is defined as follows:  
       
         
           
             
               
                 
                   T 
                   0 
                   l 
                 
                 = 
                 2 
               
               , 
               
                 
                   T 
                   1 
                   l 
                 
                 = 
                 
                   
                     l 
                      
                     
                         
                     
                      
                     and 
                      
                     
                         
                     
                      
                     
                       T 
                       
                         n 
                         + 
                         1 
                       
                       l 
                     
                   
                   = 
                   
                     
                       l 
                       · 
                       
                         T 
                         n 
                         l 
                       
                     
                     - 
                     
                       T 
                       
                         n 
                         - 
                         1 
                       
                       l 
                     
                   
                 
               
               , 
             
           
           
           
               
           
         
         where the subscript denotes the nth term while the superscript denotes the order l.  
       
     
     
         8 . A random number generator, comprising: 
 means for determining a seed number;    means for forming at least one of sums and differences of the seed number and a series of prime numbers; and    means for outputting last digits of the series of prime numbers to produce a set of random digits.