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Search: id:A120210
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| A120210 |
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Integer squares y from the smallest solutions of y^2 = x*(a^N - x)*( b^N + x) (elliptic line, Weierstrass equation) with a and b legs in primitive Pythagorean triangles and N = 2. Sequence ordered in increasing values of leg a. |
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+0
4
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| 20, 30, 156, 600, 420, 1640, 3660, 520, 2590, 7140, 1224, 10920, 8190, 20880, 32580, 4872, 19998, 5220, 48620, 69960, 3150, 41470, 97656, 132860, 19080, 76830, 176820, 230880, 131070, 12740, 296480, 11100, 375156, 52360, 209950, 468540, 64080
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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The case x congruent to 0 mod b or b congruent to 0 mod x is frequent (e.g. A120212). Note that the triads a = 3 b = 4 c = 5 and a = 4 b = 3 c = 5 provide a different result for (x, y).
The natural solution is y = c * b * (c-b) and x = b * (c-b) with c hypotenuse in the triad - Giorgio Balzarotti (greenblue(AT)tiscali.it), Jul 19 2006
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EXAMPLE
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First primitive Pythagorean triad: 3, 4, 5
Weierstrass equation. y^2 = x*( 3^2 -x)*( 4^2 + x)
Smallest integer solution (x, y) = (4,20)
First element in the sequence y = 20
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MAPLE
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flag :=1; x:=0; # a, b, c primitive Pythagorean triad while flag =1 do x:=x+1; y2:= x*( a^2 - x)*(x+b^2); if ((floor(sqrt(y2)))^2=y2)then print( sqrt(y2)); flag :=0; fi; od;
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CROSSREFS
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Cf. A009003, A020884, A120211, A120212, A120213.
Sequence in context: A142342 A008444 A066214 this_sequence A166631 A167360 A113760
Adjacent sequences: A120207 A120208 A120209 this_sequence A120211 A120212 A120213
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KEYWORD
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nonn
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AUTHOR
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Giorgio Balzarotti, Paolo P. Lava (greenblue(AT)tiscali.it), Jun 10 2006
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