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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