System for Finding Integer Solutions

Abstract

A system and a method for finding integer solutions of equations whose graphs are conic sections. The system provides a physical implementation of a geometric formulation of integer solutions of conic sections. The system includes a first source of waves and an arrangement of a plurality of reflectors to provide a lattice of interference patterns of standing waves in a plane, the lattice representing intersections at integer values. The system also includes a second source of waves and a detector provided along a curve that, with the second source, defines a cone of waves, which intersects with the plane of the lattice to provide a conic section. The detector finds points of intersection of the lattice and the conic section to determine integer solutions of the conic section. The conic section may be y=N/x, in which case the integer solutions provide a factorization into integers of N.

1 . A system for providing a physical implementation of a geometric formulation of integer solutions of conic sections, comprising:
 a first source of waves;   an arrangement of a plurality of reflectors to provide a lattice of interference patterns of said waves in a plane;   a second source of waves; and   a detector provided along a curve that, with the second source of waves, defines a cone that intersects with the plane of the lattice to provide a conic section, and detects points of intersection of the lattice and the conic section to determine an integer solution of the conic section.   
   
   
       2 . The system of  claim 1 , further comprising a conic equation to determine the integer solution of the conic section. 
   
   
       3 . The system of  claim 1 , wherein the first source of waves is a source of transverse waves. 
   
   
       4 . The system of  claim 1 , wherein the conic section is part of the graph of y=N/x and the integer solution provides a factorization into N, where N is an integer. 
   
   
       5 . The system of in  claim 4 , wherein N is changed by varying a wavelength of the first source of waves, varying a position of the second source of waves, and varying a position of the detector. 
   
   
       6 . The system of  claim 5 , wherein the wavelength of the first source is provided as 2/N units, where a unit is defined as a unit of length for the system. 
   
   
       7 . The system of  claim 5 , wherein the position of the second source of waves and the position of the detector are varied in a direction perpendicular to the plane of the lattice so as to be distant from a plane by 
     
       
         
           
             
               2 
               N 
             
           
         
       
     
     units, where a unit is defined as a unit of length for the system. 
   
   
       8 . The system of  claims 4 , wherein the detector detects points on the detector where radiation from the second source is weakest. 
   
   
       9 . The system of  claim 8 , wherein if the lattice plane is in the x,y-plane and a detected point has an x-coordinate of a units, then 
     
       
         
           
             
               
                 Na 
                 
                   2 
                   - 
                   a 
                 
               
             
              
             
               
                   
               
                
               
                   
               
             
              
             and 
              
             
                 
             
              
             
               
                 
                   N 
                    
                   
                     ( 
                     
                       2 
                       - 
                       a 
                     
                     ) 
                   
                 
                 a 
               
             
           
         
       
     
     are factors of N. 
   
   
       10 . The system of  claim 1 , wherein the first and second sources of waves are sources of electromagnetic waves, a plurality of reflectors is in the form of a plurality of mirrors, and the detector is in the form of an electromagnetic sensor. 
   
   
       11 . The system of  claim 10 , wherein the plurality of mirrors have a height as small as possible while maintaining sufficient physical strength. 
   
   
       12 . The system of  claim 10 , wherein a variable resistor is provided to vary the wavelength of the first source of waves to scan through a range of values of wavelength. 
   
   
       13 . The system of  claim 1 , wherein the first source of waves generates waves on the surface of a liquid and the second source of waves is a source of visible light. 
   
   
       14 . The system of  claim 13 , wherein the detector is submerged in the liquid and detects visible light to determine points of intersection of the lattice and the conic section. 
   
   
       15 . A system for providing a physical implementation of a geometric formulation of integer solutions of conic sections, comprising:
 a first source of waves;   an arrangement of a plurality of reflectors to provide a lattice of interference patterns of said waves in a plane;   a second source of waves; and   a detector provided along a curve that, with the second source of waves, defines a cone that intersects with the plane of the lattice to provide a conic section, and detects points of intersection of the lattice and the conic section to determine an integer solution of the conic section, wherein the conic section is part of the curve y=N/x and the integer solution provides a factorization into N, wherein N is an integer.   
   
   
       16 . The system of  claim 15 , wherein the first source of waves is a source of transverse waves. 
   
   
       17 . The system of  claim 15 , wherein the first and second source of waves are sources of electromagnetic waves, the plurality of reflectors is in the form of a plurality of mirrors, and the detector is in the form of an electromagnetic sensor. 
   
   
       18 . The system of  claim 17 , wherein the plurality of mirrors have a height as small as possible while maintaining sufficient physical strength. 
   
   
       19 . The system of  claim 17 , further comprising a variable resistor to vary the wavelength of the first source of waves. 
   
   
       20 . The system of  claim 19 , wherein said variable resistor scans through a range of values of wavelength.