Link between Pythagoras’
and Beal’s equation
It is found that there is a definite correspondence
between the Pythagoras’ equation and
Beal’s equation .
Let a2 + b2 = c2 be a known Pythagoras’ equation. With
simple mathematical manipulation, it can be converted into a
multi-power relation px + qy = rz
which reveals the basic
characteristics of Beal’s conjecture. It states that if p,q,r,x,y and z are
positive integers with (x,y,z) > 2 then p,q,and r have a common prime
factor.
The conversion of a Pythagoras’ equation into a Beal’s
equation can be done with fundamentals of Algebra . It is exemplified as
follows.
a2 + b2 = c2 à 1 + (b2/a2) = (c2/a2)
Multiply by pn à pn + pn (b2/a2)
= pn(c2/a2)
If b2/a2 = qn then c2/a2
= qn + 1 . If p = qn + 1 we get multi-power relation with exponents
[n,n,n+1]
(qn +1)n + [q(qn + 1)]n = (qn + 1)n+1
n=2, q =2 gives 52
+ 102 = 53
n=3,q=2 gives 93 + 183 = 94
n=3 , q=3 gives 283 + 843 = 284
if b2/a2 = qn-m/p then c2/a2
= [ 1 + qn-m/p]
pn +
(pq)n-1 = pn-1 [p + q n-1] =
(pr)n-1 where p = rn-1 –
qn-1, we get multi-power relation with exponents [n,n-1,n-1]
(rn-1 – qn-1 )n + [q (rn-1
– qn-1 )]n-1 = [r(rn-1 – qn-1
)]n-1
When n= 4, r=5,q=4
; 614 + 2443 = 3053
When n=5 ,r=4 ,q=2;
2405 + 4804 =
9604
If b2/a2
=
p/αn+1 then c2 /a2 = [ 1 + p/αn+1] = βn
p = αn+1[βn – 1], we get
multi-power relation with exponents [n,n+1,n)
when n = 3, α = 2,β
= 3; p = 416 and 4163 + 2084 = 12483
when n = 3 α = 1,β
=2; p = 7 and 73 + 74 = 143
when n = 4 , α = 2,β
=3; p =2560 and 25604 + 12805 = 76804
