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10-12-09, 11:12
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Registered User
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Join Date: Jul 2009
Location: NY
Posts: 886
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Quote:
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Originally Posted by DB2Plus
This is for performance and removing duplicates ?
Kara S.
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Yes, it is !
Lenny
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10-14-09, 17:59
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Registered User
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Join Date: Jul 2009
Location: NY
Posts: 886
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Generalized Fermat's Theorem -- experiment
Generalized Fermat's Theorem:
Code:
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
We can make some experiment:
For eps = 1e-6 we have 17 solutions in this interval....
But for eps = 1e-7 we have only one solution:
Code:
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
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10-15-09, 11:20
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Registered User
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Join Date: Jul 2009
Location: NY
Posts: 886
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more interesting results
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Quote:
242^3 + 720^3 = 729^3 - 1
244^3 + 729^3 = 738^3 + 1
334^3 + 438^3 = 495^3 - 1
372^3 + 426^3 = 505^3 - 1
426^3 + 486^3 = 577^3 - 1
566^3 + 833^3 = 904^3 - 1
791^3 + 812^3 = 1010^3 - 1
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etc... 
All you can get using GFT with lim = 1000, n = 3, eps = 1E-8
Code:
with pN(n, lim, eps) as
( select 3, 1000, double(1e-8) from sysibm.sysdummy1)....
Lenny
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Last edited by Lenny77; 10-19-09 at 15:07.
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10-19-09, 15:23
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Registered User
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Join Date: Jul 2009
Location: NY
Posts: 886
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N = 4
For N = 4 (X^4 + Y^4 = Z^4) is not such nice:
Code:
with pN(n, lim, eps) as
( select 4, 700, double(1e-7) 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 "..X.." , Y "..Y..", Z "..Z.." , 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
Result:
Quote:
..X.. ..Y.. ..Z.. Power Absolute Difference Relative Difference
167 192 215 4 -1.92000000000000E+002 -8.98560555129269E-008
242 471 479 4 1.10400000000000E+003 2.09713808541068E-008
334 384 430 4 -3.07200000000000E+003 -8.98560555129269E-008
216 647 649 4 5.18400000000000E+003 2.92204040963640E-008
179 635 636 4 -1.22900000000000E+004 -7.51144320215724E-008
501 576 645 4 -1.55520000000000E+004 -8.98560555129269E-008
452 602 645 4 1.69930000000000E+004 9.81818568668302E-008
496 627 681 4 -1.77760000000000E+004 -8.26505138634692E-008
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For eps = 1e-9 we don't have any solutions for this interval.
Lenny
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10-20-09, 14:26
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Registered User
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Join Date: Jul 2009
Location: NY
Posts: 886
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N = 5
Code:
with pN(n, lim, eps) as
( select 5, 1000, double(1e-8) 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 "..X.." , Y "..Y..", Z "..Z.." , 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
Result:
Quote:
..X.. ..Y.. ..Z.. Power Absolute Difference Relative Difference
494 954 961 5 2.43155300000000E+006 2.96665314536426E-009
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So far, so bad !
Lenny
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10-26-09, 13:06
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Registered User
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Join Date: Jul 2009
Location: NY
Posts: 886
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modification to Fermat's solution
I have made some modification to original Fermat's solution:
1. Remove all solutions where GCD(X, Y) > 1
for this I add to query:
Code:
Divisors(div) as
(select int(2) from sysibm.sysdummy1
union all
(select div + 1 from Divisors, pn
where div <= sqrt(start + lim) * power(2, 1. / n) + 1))
.....
.....
.....
and not exists
(select 1 from Divisors F
where mod(X.A, F.div) = 0
and mod(Y.A, F.div) = 0
and X.A > F.div
and Y.A > F.div )
)
2. Get solutions not in interval [0, lim], but in interval [start, start + lim]
Quote:
2020 2121 2929 2020^2 + 2121^2 = 2929^2
2059 2100 2941 2059^2 + 2100^2 = 2941^2
2060 2163 2987 2060^2 + 2163^2 = 2987^2
2040 2201 3001 2040^2 + 2201^2 = 3001^2
2117 2244 3085 2117^2 + 2244^2 = 3085^2
2140 2247 3103 2140^2 + 2247^2 = 3103^2
2184 2263 3145 2184^2 + 2263^2 = 3145^2
2180 2289 3161 2180^2 + 2289^2 = 3161^2
2120 2409 3209 2120^2 + 2409^2 = 3209^2
2175 2392 3233 2175^2 + 2392^2 = 3233^2
2260 2373 3277 2260^2 + 2373^2 = 3277^2
2325 2332 3293 2325^2 + 2332^2 = 3293^2
2387 2484 3445 2387^2 + 2484^2 = 3445^2
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Query:
Code:
with pN(n, lim, start) as
( select 2, 500, 2009 from sysibm.sysdummy1)
,
Divisors(div) as
(select int(2) from sysibm.sysdummy1
union all
(select div + 1 from Divisors, pn
where div <= sqrt(start + lim) * power(2, 1. / n) + 1)
)
,
Arguments(A, ApN, start, n, lim) as
(select decimal(start, 31, 0), decimal(power(start, N), 31, 0),
decimal(start, 31, 0), n, lim
from pN
union all
select A + 1, power(A + 1, N), start, n, lim
from Arguments
where A + 1 <= (start + lim) * power(2, 1. / n) + 1
)
select X.A x, Y.A y, Z.A z,
replace(
varchar(X.A) || '^' || varchar(x.N) || ' + ' ||
varchar(Y.A) || '^' || varchar(y.N) || ' = ' ||
varchar(Z.A) || '^' || varchar(z.N), '.', '' ) "Fermat Solution"
from Arguments X, Arguments Y, Arguments Z
where
X.ApN + Y.ApN = Z.ApN
and Y.A > X.A
and Z.A > Y.A
and X.A + Y.A > Z.A
and X.A <= x.start + x.lim
and Y.A <= y.start + y.lim
and not exists
(select 1 from Divisors F
where mod(X.A, F.div) = 0
and mod(Y.A, F.div) = 0
and X.A > F.div
and Y.A > F.div )
order by Z.A, X.A, Y.A
Lenny Khiger, ADSPA&VP
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Fermat's Theorem solution (plain DB2)
