1,1

For a primitive Pythagorean triangle with sides X, Y & Z, we have two generating numbers m&n such that m>n, gcd(m,n) = 1 and the parity of m&n are opposite. X = m^2 - n^2, Y = 2mn, and Z = m^2 + n^2, s = m^2 + mn and finally r = n(m-n).

Moreover, a primitive Pythagorean triangle has area n*a(n).

Albert H. Beiler, "Recreations In The Theory Of Numbers, The Queen Of Mathematics Entertains," Dover Publications, Inc., Second Edition, NY, 1966, Chapter XIV, 'The Eternal Triangle,' pages 104 - 134.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics

Wm. H. Richardson, The inradius of a Right Triangle with Integral Sides

When n is i)an odd prime power, s = (n + 1)(n + 2). ii)a power of 2, s = (n + 1)(2n + 1). iii)a composite with relatively prime factors a*b such that a is smallest, s = (a + b)(2a + b).

a = Table[10^9, {75} ]; Do[ If[ GCD[m, n] == 1 && Sort[ Mod[ {m, n}, 2]] == {0, 1}, s = m^2 + m*n; r = n(m - n); If[r < 76 && a[[r]] > s, a[[r]] = s; Print[r, " ", s]]], {m, 2, 10^2}, {n, 1, m - 1} ]

Cf. A014498.

Sequence in context: A020886 A093508 A094183 this_sequence A012412 A009092 A015793

Adjacent sequences: A056898 A056899 A056900 this_sequence A056902 A056903 A056904

nonn

Lekraj Beedassy (blekraj(AT)yahoo.com), Feb 12 2002

Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 18 2002