[lennykhiger]

Fermat's Theorem X^n + Y^n = Z^n where X, Y, Z, n are integer numbers and n >= 2.

I created the simple query which give you ability to play with this therem.

with pN(n, lim) as
( select 2, 500 from sysibm.sysdummy1)
,
ArgX(x, XpN) as 
(select int(1), int(1) from sysibm.sysdummy1
union all
select X + 1, power(X + 1, N)
from ArgX, pN
where X + 1 <= lim
) 
,
ArgY(y, YpN) as 
(select * from ArgX )
,
ArgZ(z, ZpN) as 
(select int(1), int(1) from sysibm.sysdummy1
union all
select Z + 1, power(Z + 1, N)
from ArgZ, pN
where Z + 1 <= lim * power(2, 1. / n) + 1
) 
select X, Y, Z, 
strip(digits(X), L, '0') || '^' || strip(digits(N), L, '0') || ' + ' ||
strip(digits(Y), L, '0') || '^' || strip(digits(N), L, '0') || ' = ' ||
strip(digits(Z), L, '0') || '^' || strip(digits(N), L, '0') "Fermat Solution"

from ArgX, ArgY, ArgZ, pN 

where 
     ZpN = XpN + YpN
and X < Y
and Y < Z
and Z < X + Y 

order by Z, X, Y

With this query you can easily get the first 500 (actually 386) arguments which are the solution for Fermat Great theorem for N = 2.

You can change N in pN table and you'll not find any argument which will
satisfy the Fermat's equation (if N >=3).

Lenny Khiger, ADSPA&VP

[lennykhiger]

Generalized Fermat's Theorem:

abs(((X^n + Y^n) / Z^n) - 1.0) < eps,
where 0 < X < Y < Z

with pN(n, lim, eps) as
( select 3, 300, double(1e-6) from sysibm.sysdummy1)
,
ArgX(x, XpN) as 
(select int(1), double(1) from sysibm.sysdummy1
union all
select X + 1, power(double(X + 1), N)
from ArgX, pN
where X + 1 <= lim
) 
,
ArgY(y, YpN) as 
(select * from ArgX )
,
ArgZ(z, ZpN) as 
(select int(1), double(1) from sysibm.sysdummy1
union all
select Z + 1, power(double(Z + 1), N)
from ArgZ, pN
where Z + 1 <= lim * power(2, 1. / n) + 1
) 
select X argX, Y argY, Z argZ , N "Power", (ZpN - (XpN + YpN)) "Absolute Difference" , 
double(ZpN) / double(XpN + YpN) - 1.0 "Relative Difference" 

from ArgX, ArgY, ArgZ, pN 
where ZpN between (XpN + YpN) * ( 1. - eps) and (XpN + YpN) * ( 1. + eps) 
and X < Y
and Y < Z
and X < Z
and Z <= X + Y 
order by abs(ZpN - (XpN + YpN)), X, Y

For eps = 1e-6 we have 17 solutions in this interval....
But for eps = 1e-7 we have only one solution:

select power(135, 3) + power(235, 3) - power(249, 3) 
from sysibm.sysdummy1

We can see 135^3 + 235^3 = 248^3 + 1 (not bad !).

For eps = 1e-8 we have no solutions in this interval, even for Generalized
Fermat's Theorem.

Maybe that's why Fermat's Theorem doesn't have solutions when n >=3.

Lenny Khiger, ADSPA&VP