Beal's conjecture

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  p^x  + q^y  = r^z  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  = q^n-m/p then c²/a² = [ 1 + q^n-m/p]

pⁿ  + (pq)^n-1    = p^n-1 [p + q ^n-1] = (pr)^n-1  where p = r^n-1 – q^n-1, we get multi-power relation with exponents [n,n-1,n-1]

(r^n-1 – q^n-1 )ⁿ  +  [q (r^n-1 – q^n-1 )]^n-1 = [r(r^n-1 – q^n-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