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A270558 Area of the smallest convex n-gon such that all its sides, diagonals and area are distinct integers. 0

%I

%S 6,378,9870,196950,5695998,12473070

%N Area of the smallest convex n-gon such that all its sides, diagonals and area are distinct integers.

%C Does such an n-gon exist for all n >= 3? - _Dmitry Kamenetsky_, Feb 15 2017

%H Carlos Rivera, <a href="http://www.primepuzzles.net/problems/prob_064.htm">Problem 64: prime convex quadrilateral</a>.

%H Carlos Rivera, <a href="http://www.primepuzzles.net/problems/prob_065.htm">Problem 65: prime convex polygons</a>.

%e For n=3, we have the triangle with sides 3, 4 and 5. Its area is 6.

%e For n=4, we have the quadrilateral with sides 10, 17, 28, 35, diagonals 21 and 39, and area 378.

%e For n=5, we have the 5-gon with sides 21, 41, 175, 140, 85, diagonals 50, 105, 104, 204, 195, and area 9870.

%e For n=6, we have the 6-gon with sides 47, 663, 264, 169, 105, 1020, diagonals 700, 884, 975, 855, 952, 1001, 425, 520, 272, and area 196950.

%e For n=7, we have the 7-gon with sides 235, 1320, 1360, 2340, 612, 525, 5100, diagonals 1547, 2805, 4557, 4875, 2600, 4420, 4760, 5005, 3500, 3952, 4301, 2880, 3315, 1131, and area 5695998.

%e For n=8, we have the 8-gon with sides 1547, 612, 525, 5355, 235, 4275, 845, 1360, diagonals 2125, 2600, 5365, 5304, 2163, 1131, 5520, 5525, 3500, 2805, 5460, 5491, 3952, 3315, 5408, 4301, 3720, 4420, 4875, 4760, and area 12473070.

%e See the Rivera links for images of these.

%K nonn,more

%O 3,1

%A _Dmitry Kamenetsky_, Mar 18 2016

%E a(6)-a(8) from _Luca Petrone_, Feb 06 2017

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