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A111200 Solution of Diophantine equation (1/x^2 + 1/y^2 = 1/z^2); x values in increasing order. 2
15, 20, 30, 40, 45, 60, 65, 75, 80, 90, 100, 105, 120, 130, 135, 136, 140, 150, 156, 160, 165, 175, 180, 195, 200, 210, 220, 225, 240, 255, 260, 270, 272, 280, 285, 300, 312, 315, 320, 325, 330, 340, 345, 350, 360, 369, 375, 380, 390, 400, 405, 408, 420, 435 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Similar to integral Pythagorean triangles, but with reciprocal of integers: 1/x and 1/y are the legs and 1/z is the hypotenuse. Properties:

1) x*y is congruent to 0 mod z;

2) x = u*v *(u^2+v^2) * m /2 or x = (u^2-v^2)*(u^2+v^2)*m/4; y = (u^2-v^2)*(u^2+v^2)*m/4 or y = u*v *(u^2+v^2)*m /2; z = (u^2-v^2)*u*v*m /2; u and v are odd and relatively prime, u > v, m is an integer.

To use the equations: set n, evaluate (2*n^2+2*n+1), derive m = (2*n^2+2*n+1)*integer, then evaluate x, y, z.

3) Primitive integral solution, i.e., those solutions in which there is no factor common to x, y and z, are (15, 20, 12), (65, 156, 60), (136,255,120), (175, 600, 168), (369,1640, 360), (671, 3660, 660), ...

Primitive integral solution are those where m = 1.

4) If the triad (a, b, c) is a solution of a Pythagorean triangle, i.e., a^2 + b^2 = c^2, then (x = a*c, y = b*c, z= a*b) is a solution of (1/x^2 + 1/y^2 = 1/z^2). For Pythagorean triads see for example http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html [broken link]

5) First multiple solutions (more values for y and z given x ) are x = 60,120,180,195. Multiple solution can be derived by equations at point 2.

6) No prime is in the sequence.

Related topics: the Diophantine equation (1/x^i + 1/y^i = 1/z^i) has no integral solutions for i>2, as it is easy to demonstrate by means of the Fermat-Wiles theorem.

LINKS

Table of n, a(n) for n=1..54.

EXAMPLE

a(1) = 15 because 1/15^2 + 1/20^2 = 1/12^2 (smallest solution); a(2)= 20 as 1/20^2 + 1/15^2 = 1/12^2.

MAPLE

recPy:=proc(n) local x, y, z, Rx2, Ry2; for x from 1 by 1 to n do Rx2:=1/x^2; for y from 1 by 1 to x^2 do Ry2:=1/y^2; for z from 1 by 1 to x do if (Rx2 + Ry2 =1/z^2) then print(x); fi; od; od; od; end: # convert into set

CROSSREFS

Cf. A094807.

Sequence in context: A163602 A074236 A086770 * A088494 A109659 A294149

Adjacent sequences:  A111197 A111198 A111199 * A111201 A111202 A111203

KEYWORD

nonn

AUTHOR

Giorgio Balzarotti, Paolo P. Lava, Oct 24 2005

EXTENSIONS

More terms from Vladeta Jovovic, Oct 25 2005

STATUS

approved

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Last modified March 30 09:41 EDT 2020. Contains 333125 sequences. (Running on oeis4.)