A177021 Numbers which are the area of exactly three Pythagorean triangles.

840, 3360, 7560, 10920, 13440, 21000, 30240, 31920, 41160, 43680, 53760, 68040, 84000, 98280, 101640, 120960, 127680, 141960, 164640, 166320, 174720, 189000, 215040, 242760, 272160, 273000, 286440, 287280, 303240, 336000, 370440, 393120, 406560, 444360

Offset: 1

Author

Claudio Meller, on a suggestion by Antonio Roldán, Dec 08 2010

Keywords

• nonn

Comments

The triangles need not be primitive. Number of terms less than 10^n: 0, 0, 1, 3, 14, 53, ....

13123110 is the smallest number which is the area of three primitive Pythagorean triangles, (1380,19019,19069)(3059,8580,9109) and (4485,5852,7373); this triple was found by Charles L. Shedd in 1945.

From Sture Sjöstedt, Dec 06 2016: (Start)

840 = 3*5*7*8; p=3, q=8, q-p=5, r=7 is a solution to p^2 - pq + q^2 = r^2. If r is a prime number in the sequence 7, 13, 19, ..., there are three Pythagorean triangles with the same area and at least one of them is primitive.

10920 = 7*8*13*15; p=7, q=15, q-p=8, r=13.

x^2 + 3*y^2 = 4*r^2 where r is a prime number in the sequence 7, 13, 19, ... gives lattice points that can be used to find solutions to p^2 - pq + q^2 = r^2. p, q, (q-p) and r are the y-coordinates in the first quadrant. (End)

Examples

			a(1) = 840 is the area of {15,112,113}, {24,70,74} & {40,42,58}.
a(2) = 3360 is the area of {30,224,226}, {48,140,148} & {80,84,116}.
a(3) = 7560 is the area of {45,336,339}, {72,210,222} & {120,126,174}.
		

References

• Morton Cohen, Charles Lutwidge Dodgson (Lewis Carroll), b. Jan. 27, 1832, d. Jan. 14, 1898, A Brief Biography, Vintage Books, ISBN 978-0-679-74562-4 (26 November 1996).

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Pythagorean Triple.
• Index entries related to Pythagorean Triples.

Crossrefs

Cf. A009112, A177063.

Programs

Mathematica

lst = {}; m = 2; While[ m < 10^3, n = 1; While[ n < m, If[ IntegerQ@ Sqrt[ m^2 + n^2], a = m*n/2; If[a < 10^6, AppendTo[ lst, a], n = m]]; n++ ]; m++ ]; Union@ Flatten@ Select[ Split@ Sort@ lst, Length@ # == 3 &]

Formula

A177021 = { n | A177063(n)=3 }. - M. F. Hasler, Dec 09 2010

Extensions

Extended and edited by Robert G. Wilson v, Dec 08 2010

a(28)-a(34) from Giovanni Resta, Aug 16 2017