Sunday, January 17, 2016

Fun With Numbers

Fun With Numbers
Irreducible relations with equal sum of two squares can be generated from the following general expression
(P02P04 + P01P03)2  + (P01P04  –  P02P03)2  =  (P02P04  --  P01P03)2  + (P01P04  +  P02P03)2
Where P01,P02,P03,P04 are any four prime numbers. Compound numbers may also be taken, but sometimes it gives reducible set.
Interchange of any pair of prime numbers will not affect the balanced state of the relation. The equality is still preserved when all the prime numbers are expressed with same power
(P022P042 + P012P032)2 + (P012P042 – P022P032)2 = (P022P042 - P012P032)2 + (P012P042 + P022P032)2
(P02nP04n + P01n P03n)2 + ( P01nP04n  -- P02nP03n)2 = (P02nP04n - P01n P03n)2 + ( P01nP04n  + P02nP03n)2
When P01 = 1,P02= 2,P03= 3,P04=5; 132 + 11 = 72 + 112
            P01 = 3,P02= 2,P03= 1,P04=5; 132 + 132 = 72 + 172
                P01 = 3,P02= 1,P03= 5,P04=2 ; 172 + 12 = 132 + 112
While the relation with squares of prime numbers gives,
When P01 = 1,P02= 2,P03= 3,P04=5;  1092 + 112 = 912 + 612
            P01 = 3,P02= 2,P03= 1,P04=5; 1092 + 2212 = 912 + 2292

           P01 = 3,P02= 1,P03= 5,P04=2 ;2292 + 112 = 2212 + 612

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