28.4k-gonal triples determined by a Pythagorean triple 729
(p, q, r)corresponds toat most oneline onSassociated withk-gonal
triples (fork≥ 5 ).
Indeed, ifk=4h+2,(k−2)′is the even number 2 h, and cannot
divide the odd integerm−n. It follows that only those pairs(m, n), with
na multiple of 2 hgive(4h+2)-gonal pairs. For example, by choosing
m=2h+1,n=2h,wehave
p=4h+1,q=8h^2 +4h, r=8h^2 +4h+1,
a 0 =4h+1,b 0 =8h^2 +2h+1,c 0 =8h^2 +2h+2.These give an infinite family of(4h+2)-gonal triples:
at =(4h+1)(t+1),
bt =8h^2 +2h+1+(8h^2 +4h)t,
ct =8h^2 +2h+2+(8h^2 +4h+1)t.(4h+2)−gonal triples(h, k, g) (m, n) (p, q, r) (a, b, c)
(1, 6 ,4) (3,2) (5, 12 ,13) (5, 11 ,12)
(5,2) (21, 20 ,29) (14, 13 ,19)
(5,4) (9, 40 ,41) (9, 38 ,39)
(7,2) (45, 28 ,53) (18, 11 ,21)
(7,4) (33, 56 ,65) (11, 18 ,21)
(7,6) (13, 84 ,85) (13, 81 ,82)
(9,2) (77, 36 ,85) (11, 5 ,12)
(9,4) (65, 72 ,97) (13, 14 ,19)
(9,8) (17, 144 ,145) (17, 140 ,141)
(11,2) (117, 44 ,125) (104, 39 ,111)
(11,4) (105, 88 ,137) (60, 50 ,78)
(11,6) (85, 132 ,157) (68, 105 ,125)
(11,8) (57, 176 ,185) (38, 116 ,122)
(11,10) (21, 220 ,221) (21, 215 ,216)
(2, 10 ,^83 ) (5,4) (9, 40 ,41) (9, 37 ,38)
(7,4) (33, 56 ,65) (33, 55 ,64)
(9,4) (65, 72 ,97) (52, 57 ,77)
(9,8) (17, 144 ,145) (17, 138 ,139)
(11,4) (105, 88 ,137) (90, 75 ,117)
(11,8) (57, 176 ,185) (57, 174 ,183)
(3, 14 ,^125 ) (7,6) (13, 84 ,85) (13, 79 ,80)
(11,6) (85, 132 ,157) (85, 131 ,156)