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A177021
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Numbers which are the area of exactly three Pythagorean triangles.
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4
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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
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OFFSET
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1,1
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COMMENTS
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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.
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)
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REFERENCES
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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).
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LINKS
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FORMULA
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EXAMPLE
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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}.
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MATHEMATICA
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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 &]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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