730 Polygonal triples
28.5 Correspondence between(2h+1)-gonal and 4 h-
gonal triples
Letk 1 <k 2 be two positive integers≥ 5. Theorem 2 suggests that there
is a one-to-one correspondence betweenk 1 -gonal triples andk 2 -gonal
triples, provided(k 1 −2)′=(k 2 −2)′. This is the case if and only if
k 1 =2h+1,k 2 =4h,for someh≥ 2. (28.12)In this case,(k 1 −2)′=(k 2 −2)′=2h− 1 , while(k 1 −4)′=2h− 3 ,
and(k 2 −4)′=2h− 2. The(2h+1)-gonal triple(a, b, c)and a 4 h-gonal
triple(a′,b′,c′)are related by
(m−n)(c−c′)≡n
2 h− 1(modm^2 +n^2 ).(2h+1)−gonal and 4h−gonal triples(2h+1)−gonal 4 h−gonal
(h, 2 h+1, 4 h) (m, n) (p, q, r) (a, b, c) (a′,b′,c′)
(2, 5 ,8) (4,1) (15, 8 ,17) (7, 4 ,8) (14, 8 ,16)
(4,3) (7, 24 ,25) (7, 23 ,24) (7, 22 ,23)
(5,2) (21, 20 ,29) (5, 5 ,7) (10, 10 ,14)
(7,4) (33, 56 ,65) (4, 7 ,8) (8, 14 ,16)
(7,6) (13, 84 ,85) (13, 82 ,83) (13, 80 ,81)
(8,3) (55, 48 ,73) (22, 19 ,29) (44, 38 ,58)
(8,5) (39, 80 ,89) (35, 72 ,80) (31, 64 ,71)
(10,1) (99, 20 ,101) (48, 10 ,49) (96, 20 ,98)
(10,3) (91, 60 ,109) (26, 17 ,31) (52, 34 ,62)
(10,7) (51, 140 ,149) (40, 110 ,117) (29, 80 ,85)
(10,9) (19, 180 ,181) (19, 177 ,178) (19, 174 ,175)
(3, 7 ,12) (6,1) (35, 12 ,37) (16, 6 ,17) (33, 12 ,35)
(6,5) (11, 60 ,61) (11, 57 ,58) (11, 56 ,57)
(7,2) (45, 28 ,53) (33, 21 ,39) (44, 28 ,52)
(8,3) (55, 48 ,73) (27, 24 ,36) (36, 32 ,48)
(8,5) (39, 80 ,89) (39, 79 ,88) (26, 52 ,58)
(9,4) (65, 72 ,97) (24, 27 ,36) (32, 36 ,48)
(4, 9 ,16) (8,1) (63, 16 ,65) (29, 8 ,30) (60, 16 ,62)
(8,7) (15, 112 ,113) (15, 107 ,108) (15, 106 ,107)
(9,2) (77, 36 ,85) (18, 9 ,20) (37, 18 ,41)
(10,3) (91, 60 ,109) (75, 50 ,90) (90, 60 ,108)
(10,7) (51, 140 ,149) (17, 45 ,48) (51, 138 ,147)
(5, 11 ,20) (10,1) (99, 20 ,101) (46, 10 ,47) (95, 20 ,97)
(10,9) (19, 180 ,181) (19, 173 ,174) (19, 172 ,173)