OFFSET
1,1
COMMENTS
All primitive Pythagorean triples (see the links) (a,b,c), with a odd, b even, hence c odd, are given by c=u^2 + v^2, with u odd, u=2*n+1, n>=1, v even, v=2*m, m>=1, and gcd(u,v)=1. The present array is c=c(n,m) = (2*n-1)^2 + (2*m)^2, if gcd(2*n-1,2*m)=1 and 0 otherwise. The corresponding triangle, read by SW-NE diagonals, is T(n,m):= c(n-m+1,m). The 0 entries indicate that there are only non-primitive triples for these n,m values. See the example section for the scaling factor g=gcd(u,v)^2 for such non-primitive triangles.
All primitive Pythagorean triples are given by
(a(n,m)=A208854(n,m), b(n,m)=A208855(n,m), c(n,m)), n>=1, m>=1. If this is (0,0,0) then no primitive triple exists for these n,m values. See the example section.
In the prime factorization of c(n,m) (which is odd) all prime factors are of the type 4*k+1 (see A002144). See the Niven-Zuckerman-Montgomery reference, Theorem 3.20, p. 164. For the general representation of positive integers as the sum of two squares see Theorem 2.15 by Fermat, p. 55. E.g.: c(5,2) = 85 = 5*17. c = 5*7^2 = 245 has a non-primitive solution 7^2*(1^2 + 2^2) = 7^2*c(1,1), therefore c(4,7)=0 in this array.
The triples with an even cathetus (b) and the hypotenuse (c) differing by 1 unit are (2*k+1, 4*T(k), 4*T(k)+1), k >= 1, with the triangular numbers A000217. The c values are given in A001844. E.g., (n,m)=(1,1), k=1. (3,4,5); (n,m)=(2,1), k=2, (5,12,13); (n,m)=(2,2), k=3, (7,24,25). See the example section for the table.
The triples with an odd cathetus (a) and the hypotenuse differing by 2 units are (4*k^2-1, 4*k, 4*k^2+1), k >= 1. These triples are given in (A000466(k), A008586(k), A053755(k)). E.g., (n,m)=(1,4), k=4, (63,16,65).
The triples with the catheti differing by one length unit are generated by a substitution rule for the (u,v) values starting with (1,1). See a Wolfdieter Lang comment on A001653 for this rule. - Wolfdieter Lang, Mar 08 2012
REFERENCES
I. Niven, H. S. Zuckerman and H.L. Montgomery, An Introduction to the Theory of Numbers, 5th edition, Wiley & Sons, New York, 1991
LINKS
Ron Knott, Pythagorean Triples and Online Calculators.
E. S. Rowland, Primitive Solutions to x^2 + y^2 = z^2
Eric Weisstein's World of Mathematics, Pythagorean Triple.
FORMULA
T(n,m) = c(n-m+1,m), n >= m >= 1, with c(n,m) := (2*n-1)^2 + (2*m)^2, if gcd(2*n-1, 2*m) = 1 and 0 otherwise.
EXAMPLE
Triangle T(n,m):
......m| 1 2 3 4 5 6 7 8 9 10 ...
......v| 2 4 6 8 10 12 14 16 18 20 ...
n, u
1, 1 5
2, 3 13 17
3, 5 29 25 37
4, 7 53 41 0 65
5, 9 85 65 61 73 101
6, 11 125 97 85 89 109 145
7, 13 173 137 0 113 0 0 197
8, 15 229 185 157 145 149 169 205 257
9, 17 293 241 205 185 181 193 221 265 325
10,19 365 305 0 233 221 0 0 281 0 401
...
Array c(n,m):
......m| 1 2 3 4 5 6 7 8 9 10 ...
......v| 2 4 6 8 10 12 14 16 18 20 ...
n, u
1, 1 5 17 37 65 101 145 197 257 325 401
2 3 13 25 0 73 109 0 205 265 0 409
3, 5 29 41 61 89 0 169 221 281 349 0
4, 7 53 65 85 113 149 193 0 305 373 449
5, 9 85 97 0 145 181 0 277 337 0 481
6, 11 125 137 157 185 221 265 317 377 445 521
7, 13 173 185 205 233 269 313 365 425 493 569
8, 15 229 241 0 289 0 0 421 481 0 0
9, 17 293 305 325 353 389 433 485 545 613 689
10,19 365 377 397 425 461 505 557 617 685 761
...
------------------------------------------------------------------
......m| 1 2 3 4 ...
......v| 2 4 6 8 ...
n, u
1, 1 (3,4,5) (15,8,17) (35,12,37) (63,16,65)
2, 3 (5,12,13) (7,24,25) (0,0,0) (55,48,73)
3, 5 (21,20,29) (9,40,41) (11,60,61 (39,80,89)
4, 7 (45,28,53) (33,56,65) (13,84,85) (15,112,113)
5, 9 (77,36,85) (65,72,97) (0,0,0) (17,144,145)
6, 11 (117,44,125) (105,88,137) (85,132,157) (57,176,185)
7, 13 (165,52,173) (153,104,185) (133,156,205) (105,208,233)
8, 15 (221,60,229) (209,120,241) (0,0,0) (161,240,289)
9, 17 (285,68,293) (273,136,305) (253,204,325) (225,272,353)
10,19 (357,76,365) (345,152,377) (325,228,397) (297,304,425)
...
Array continued:
......m| 5 6 7 8 ...
......v| 10 12 14 16 ...
n, u
1, 1 (99,20,101) (143,24,145) (195,28,197) (255,32,257)
2 3 (91,60,109) (0,0,0) (187,84,205) (247,96,265)
3, 5 (0,0,0) (119,120,169) (171,140,221) (231,160,281)
4, 7 (51,140,149) (95,168,193) (0,0,0) (207,224,305)
5, 9 (19,180,181) (0,0,0) (115,252,277) (175,288,337)
6, 11 (21,220,221) (23,264,265) (75,308,317) (135,352,377)
7, 13 (69,260,269) (25,312,313) (27,364,365) (87,416,425)
8, 15 (0,0,0) (0,0,0) (29,420,421) (31,480,481)
9, 17 (189,340,389) (145,408,433) (93,476,485) (33,544,545)
10,19 (261,380,461) (217,456,505) (165,532,557) (105,608,617)
...
(0,0,0) indicates that no primitive Pythagorean triangle exists for these (n,m) values. The corresponding scaled triples would be (a,b,c) = g*(a/g,b/g,c/g), with g=gcd(u,v)^2 for such non-primitive triangles. E.g., c(n,m) = c(5,3) = 0, (u,v) = (9,6), g = 3^2, (45,108,117) = 3^2*(45/9,108/9,117/9) = 9*(5,12,13). The scaling factor for the primitive triangle (5,12,13), tabulated for c(n,m)=(2,1), is here 9.
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Mar 05 2012
STATUS
approved