A208853 Array of hypotenuses of primitive Pythagorean triangles when read by SW-NE diagonals.

5, 13, 17, 29, 25, 37, 53, 41, 0, 65, 85, 65, 61, 73, 101, 125, 97, 85, 89, 109, 145, 173, 137, 0, 113, 0, 0, 197, 229, 185, 157, 145, 149, 169, 205, 257, 293, 241, 205, 185, 181, 193, 221, 265, 325, 365, 305, 0, 233, 221, 0

Offset: 1

Author

Wolfdieter Lang, Mar 05 2012

Keywords

• nonn
• easy
• tabl

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.

For the increasingly ordered c-values see A008846 (with multiplicity see A020882).

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

Examples

			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
...
------------------------------------------------------------------
Array of triples (a(n,m)=A208854,b(n,m)=A208855,c(n,m)):
......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:
Array of triples (a(n,m)=A208854,b(n,m)=A208855,c(n,m)):
......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.
		

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
• Michael Somos, Table of primitive Pythagorean triples and related parameters
• Eric Weisstein's World of Mathematics, Pythagorean Triple.

Crossrefs

Cf. A020882, A002144, A208854, A208855.

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.