A263728 Primitive Pythagorean triples (a, b, c) in lexicographic order, with a < b < c.
3, 4, 5, 5, 12, 13, 7, 24, 25, 8, 15, 17, 9, 40, 41, 11, 60, 61, 12, 35, 37, 13, 84, 85, 15, 112, 113, 16, 63, 65, 17, 144, 145, 19, 180, 181, 20, 21, 29, 20, 99, 101, 21, 220, 221, 23, 264, 265, 24, 143, 145, 25, 312, 313, 27, 364, 365, 28, 45, 53
Offset: 1
Examples
The first few triples are [3, 4, 5], [5, 12, 13], [7, 24, 25], [8, 15, 17], [9, 40, 41], [11, 60, 61], [12, 35, 37], [13, 84, 85], [15, 112, 113], [16, 63, 65], [17, 144, 145], [19, 180, 181], [20, 21, 29], [20, 99, 101], ... - _N. J. A. Sloane_, Dec 15 2015
References
- H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, Chapter 5, Section 5.3.
Links
- Colin Barker, Table of n, a(n) for n = 1..10000
- Yoshinosuke Hirakawa and Hideki Matsumura, A unique pair of triangles, Journal of Number Theory, Volume 194, January 2019, Pages 297-302. About the remarkable (135,352,377) Pythagorean triple.
- Eric Weisstein's World of Mathematics, Pythagorean Triple
- Wikipedia, Pythagorean triple
Crossrefs
Programs
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Maple
a:=[]; b:={}; M:=30; for u from 2 to M do for v from 1 to u-1 do if gcd(u,v)=1 and u+v mod 2 = 1 then t1:=u^2-v^2; t2:= 2*u*v; t3:=u^2+v^2; w:=sort([t1,t2]); a:=[op(a), [op(w),t3]]; b:={ op(b), op(w), t3}; fi: od: od: a; sort(a); # A263728 sort(b); # A016825 and A042965 (Maple code from N. J. A. Sloane, Dec 15 2015)
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PARI
\\ Primitive Pythagorean triples (a,b,c) with a
Comments