A201223 Primitive Eisenstein triples (a,b,c) listed as groups of three in order of increasing b.

3, 8, 7, 5, 8, 7, 7, 15, 13, 8, 15, 13, 5, 21, 19, 16, 21, 19, 11, 35, 31, 24, 35, 31, 7, 40, 37, 33, 40, 37, 13, 48, 43, 35, 48, 43, 16, 55, 49, 39, 55, 49, 9, 65, 61, 56, 65, 61, 32, 77, 67, 45, 77, 67, 17, 80, 73, 63, 80, 73, 40, 91, 79, 51, 91, 79, 11, 96, 91

Offset: 1

Author

John W. Layman, May 09 2012

Keywords

• nonn

Comments

An Eisenstein triple is a triple (a,b,c) of positive integers with a

Examples

			(a,b,c)=(3,8,7) is an Eisenstein triple since 3<7<8 and 3^2 - 3*8 + 8^2 = 7^2.  GCD(3,8,7) = 1, so the triple is primitive.  No Eisenstein triple exists with b<8, so a(1)=3, a(2)=8, a(3)=7.
		

Links

• Danny Rorabaugh, Table of n, a(n) for n = 1..1494
• Russell A. Gordon, Properties of Eisenstein Triples, Mathematics Magazine 85 (2012), 12-25.

Crossrefs

Cf. A121992.

Programs

Mathematica

x = {}; For[b = 1, b <= 77, b++, For[c = 1, c < b, c++, For[a = 1, a < c, a++, {If[(a^2 - a*b + b^2 == c^2) && (GCD[a, b, c] == 1), AppendTo[x, {a, b, c}]]}]]]; Flatten[x]

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