(* The exact check out of  SUM OF THE POWERS,EQUAL
TO A POWER AT INFINITY.(Powers to Power at Infinity) 
theorem ; Theorem of Elhayy  with two elements	*)
(*----------------------------------------------	*)
n=850    ;(* n being a + integer 			*) 
(* As n-->Infinity El'Hayy holds true 			*)
(*--------------------------------------------------*)
t=5 ; 	(*   t being a + integer   			*)                                           
d=1 ;	(*  d being a + or - integer (not zero)	*)
(* t>d for converging     *)
(*If d=1 then we have the famous FLT cond.    	*)	
(*-----------------------------------------------	*)
m1=2      ;   (* This part generates the seed  	*)
m2=5      ;   (* of the integers   a1,a2,  b 	*) 
mT:=m1+m2
(*  m1,m2 being  + integers (or real numbers)	*)
(*-----------------------------------------------	*)
w1:=Sum[Binomial[n,t*r-d]*(m1^r),
{r,Ceiling[d/2],Floor[(d+n)/2]}]

w2:=Sum[Binomial[n,t*r-d]*(m2^r),
{r,Ceiling[d/2],Floor[(d+n)/2]}]

w3:=Sum[Binomial[n,t*r-d]*(mT^r),
{r,Ceiling[d/2],Floor[(d+n)/2]}]

z1:=Sum[Binomial[n,t*r]*  (m1^r),{r,0,Floor[n/2]}]
z2:=Sum[Binomial[n,t*r]*  (m2^r),{r,0,Floor[n/2]}]
z3:=Sum[Binomial[n,t*r]*  (mT^r),{r,0,Floor[n/2]}]
(*---------------------------------------------	*)
an1:=z2*z3*w1
an2:=z1*z3*w2
bn:=z1*z2*w3

div:=GCD[an1,an2,bn]

a1:=an1/div
a2:=an2/div
b:=bn/div

pow:=(t/d)
Hayy2:=(a1^pow+a2^pow)/(b^pow)
N[Hayy2 ,70]

1.