Appendix C

Euler Co-incidences

To provide exhaustive lists of co-incidences would take an unreasonable amount of time, and would detract from the main aim, which is simply to locate Euler representations. However, in the course of searching, the following arise.

103 = (7,7,5,5,4)3 = (9,6,3,3,1)3

113 = (8,7,7,5,2)3 = (8,8,6,4,3)3

133 = (10,9,7,5)3 = (12,7,5,1)3

143 = (12,10,2,2,)3 = (13,8,3,2)3

153 = (11,11,7,7,3)3 = (12,9,9,5,4) = (14,7,6,4,2)3

163 = (12,12,8,4,4)3 = (14,9,7,6,4)3 = (14,10,7,2,1)3 = (15,7,7,3,2)3

173 = (12,11,10,9,5)3 = (12,12,9,8,6)3 = (12,12,11,5,1)3 = (13,11,11,3,3)3

183 = (15,12,9)3 = (16,12,2)3

223 = (16,14,14,10,4)3 = (16,16,11,10,5)3 = (16,16,12,8,6)3 = (18,14,10,9,7)3

= (18,14,12,7,1)3 = (19,11,10,9,9)3 = (19,12,11,9,1)3

263 = (20,18,14,10)3 = (21,17,15,3)3 = (21,19,11,5)3 = (24,14,10,2)3

283 = (21,19,18,18)3 = (23,19,13,9)3 = (24,20,4,4)3

= (26,15,10,1)3 = (26,16,6,4)3 = (27,13,4,2)3

353 = (30,25,5,5)3 = (34,12,11,8)3

363 = (30,24,18)3 = (32,24,4)3 (see 183)

393 = (30,27,21,15)3 = (36,21,15,3)3 = (38,15,9,7)3 = (38,16,7,2)3

- to base x = 40

In general, the longer the Euler length, the more likely it is that co-incidences arise. In the above case for 3rd powers, there is only one co-incidence (barring multiples) with Euler length 3, although the bases 6, 9, 12, 19, 20, 24, 25, 27, 29, 30, 38 and 40 all have Euler length 3.

44 = (16x2)4 = (3x3,13x1)4

84 = (16x4)4 = (3x6,13x2)4 (see preceding)

94 = (6,6,6,6,6,3)4 = (7,6,6,6,4,2)4 = (8,7,2,2,2,2)4

154 = (12,12,9,6,6)4 = (13,12,6,2,2)4 = (14,9,8,6,4)4

164 = (16x8)4 and 12 others of length 16

184 = (12,12,12,12,12,6)4 = (14,12,12,12,8,4)4 = (16,14,4,4,4,4)4 (see 94)

214 = (14,14,14,14,14,7)4 = (18,12,12,12,12,9)4

234 = (18,18,16,8,4,1)4 = (19,16,14,12,12,8)4 = (22,12,12,8,2,1)4

274 = (18,18,18,18,18,9)4 = (21,18,18,18,12,6)4 = (24,21,6,6,6,6)4 (see 94)

294 = (22,22,18,18,13,4)4 = (26,22,9,8,8,6)4

304 = (24,24,18,12,12,)4 = (26,24,12,4,4)4 = (28,18,16,12,8)4 (see 154)

324 = (16x16)4 = many others of length 16

334 = (22,22,22,22,22,11)4 = (31,16,16,16,16,4)4

354 = (28,26,22,21,4)4 = (28,28,21,14,14)4

364 = (24,24,24,24,24,12)4 = (28,24,24,24,16,8)4 = (32,28,8,8,8,8)4 (see 94)

374 = (32,29,14,14,12,12)4 = (34,22,21,18,8,2)4 = (36,21,2,2,2,2)4

394 = (26,26,26,26,26,13)4 = (36,24,21,18,6,6)4

• to base x = 40

65 = (5,4,4,4,4,3,3,2,2,1,1,1,1,1)5 = (5,5,3,3,3,3,3,3,2,2,1,1,1,1)5

205 = (16,14,14,14,14,2,2,2,2)5 = (17,17,11,11,8,5,4,4,3)5

215 = (18,15,13,13,12,12,11,7,7,3)5 = (18,18,11,10,8,6,5,3,1,1)5

225 = (17,17,16,15,13,9,9,6,6,4)5

= (17,17,17,12,11,11,11,11,4,1)5

= (18,17,15,14,12,12,8,6,6,4)5

= (20,17,13,10,9,5,3,2,2,1)5

= (21,15,12,7,7,7,6,5,1,1)5

275 = (22,22,19,15,15,8,6,5,5)5 = (23,20,20,16,13,9,7,7,2)5

= (25,19,15,14,14,12,6,6,6)5 = (26,18,14,7,7,5,5,3,2)5

285 = (22,21,21,17,17,16,3,1)5 = (26,22,10,8,8,6,4,4)5

295 = (25,23,20,14,13,10,10,2,2)5 = (26,20,19,19,13,10,5,5,2)5

= (26,21,19,18,10,9,7,5,4)5 = (28,17,15,15,10,10,10,9,5)5

315 = (28,25,15,13,12,12,7,6,3)5 = (28,25,15,14,11,11,7,7,3)5

385 = (36,27,18,18,14,10,5)5 = (37,24,18,8,5,5,1)5

- to base x = 40

665 = (50,50,47,43,40,38,37,1)5 = (55,49,45,43,42,21,11,10)5

= (57,46,42,41,41,37,26,16)5 = (58,52,42,36,30,12,8,8)5

= (59,51,36,35,32,31,28,4)5 = (61,44,38,36,35,34,22,6)5

86 = (6,6,6,6,5,5,5,5,4,4,4,3,1,1,1)6 = (6,6,6,6,6,4,4,4,4,4,4,4,2,2,2)6

116 = (9,9,8,7,7,6,6,6,6,5,4,4,3,2,1,1,1)6 = (10,9,7,7,3,3,3,3,3,3,2,2,2,2,2,2,2)6

126 = (9,9,9,9,9,6,6,6,6,6,6,6,3,3,3)6 = (9,9,9,9,9,8,6,5,4,2,2,2,2,1,1)6

= (10,9,9,9,8,6,6,5,5,4,3,2,2,1,1)6 = (10,10,9,8,6,6,6,5,5,5,4,3,3,1,1)6

136 = (11,9,9,9,9,8,8,8,7,5,4,4,3,3,2,2)6 = (11,10,9,9,8,8,8,7,6,5,5,4,4,3,3,3)6

= (11,11,9,9,7,6,6,4,4,3,3,2,2,2,1,1)6 = (11,11,10,7,6,6,6,5,4,4,3,3,3,2,1,1)6

= (12,9,9,9,7,6,6,5,5,4,2,2,2,1,1,1)6 = (12,10,9,7,6,6,6,5,5,5,4,3,2,1,1,1)6

146 = (13,10,10,9,6,6,6,5,5,2,1,1,1,1)6 = (13,11,9,8,6,6,5,5,4,4,4,3,1,1)6

156 = (12,12,11,11,11,6,5,5,5,4,4,3,3,3,2)6 = (12,12,12,11,9,6,5,5,5,5,5,4,3,2,1,)6

= (13,12,12,9,6,4,4,4,3,3,1,1,1,1,1)6 = (13,13,9,9,8,8,6,6,5,5,5,4,4,3,3)6

216 = (16,16,15,15,15,15,13,11,6,4,2,1,1,1)6 = (17,16,15,15,15,13,11,11,10,10,8,6,3,1)6

= (17,17,15,15,13,12,11,11,11,10,9,6,4,2)6 = (18,18,13,12,11,11,11,11,11,9,9,4,4,1,)6

= (20,13,12,12,11,11,11,11,11,10,9,9,6,1,)6 = (20,16,12,10,10,3,3,3,3,1,1,1,1,1)6

= (20,16,13,6,6,5,5,5,5,4,3,3,3,1)6

246 = (21,19,17,16,14,14,11,8,7,7,6,3,3)6 = (22,17,15,15,15,14,14,12,10,7,7,7,5)6

266 = (23,21,17,17,16,14,11,8,8,7,7,3)6 = (23,21,18,17,14,14,11,7,5,2,1,1)6

306 = (27,25,19,19,12,7,7,7,4,4,1)6

= (27,25,21,13,13,11,7,7,4,4,4)6

= (28,23,19,17,17,12,9,7,7,2,1)6

316 = (26,25,23,21,19,17,17,13,8,7,7)6 = (28,25,23,13,13,11,11,7,7,4,3)6

336 = (26,25,25,25,21,21,20,13,13,13,7)6

= (28,28,24,20,20,14,10,6,4,4,1)6

= (32,24,14,14,14,12,8,8,8,8,1)6

346 = (31,25,22,21,20,19,19,17,17,14,7)6

= (31,26,25,18,17,17,16,13,7,7,7)6

= (31,28,23,17,12,7,7,5,2,1,1)6

356 = (31,27,27,23,16,14,12,7,7,7,7,7,4)6 = (33,25,25,17,15,14,14,14,7,7,7,6,1)6

- to base x = 35

157 = (12,12,11,11,11,11,10,10,6,6,6,5,5,5,5,5,4,2,2,1,1)7

= (13,12,11,10,10,10,10,10,7,7,7,5,5,5,5,3,3,3,3,1,1)7

187 = (16,15,14,13,8,8,6,6,3,3,3,3,2,2)7 = (17,14,12,12,10,10,9,4,3,3,2,2,2,2)7

227 = (20,19,14,14,14,8,7,6,6,6,5,5,5,5,5,5,4)7

= (21,16,15,14,13,13,11,7,7,7,7,6,3,2,2,2,2)7

247 = (21,19,18,17,16,16,14,14,14,10,9,7,5,5,5,2)7

= (21,20,19,17,15,11,9,9,7,6,5,5,2,2,1,1)7

= (22,19,18,15,14,13,13,13,13,12,10,10,8,7,3,2)7

= (23,17,17,14,14,13,13,11,9,6,5,4,1,1,1,1)7

- to base x = 30

108 = (3x8,8x7,2x6,4,19x3,2x2,5x1)8 = (4x8,4x7,25x5,4,3x2,3x1)8

138 = (11,2x10,8x9,8,6x7,6,10x5,7x3,2,1)8

= (3x11,2x9,14x7,3x6,2x5,5x3,2x2,7x1)8

= (12,11,10,9,8,7,2x6,6x5,15x3,9x1)8

148 = (12,4x11,10,9,8,4x7,14x5,3x3,2,6x1)8 = (13,12,11,2x7,10x5,3x4,13x3,5x1)8

188 = (2x15,6x13,4x11,10,7x7,5x5,8x3,2)8 = (16,4x14,8x10,13x6,8x2)8

= (17,3x13,5x11,12x9,7,6,3x5,6x3,2,1)8 = (17,15,12,4x11,4x9,3x7,6,7x5,12x1)8

- to base x = 20

69 = (2x5,19x4,60x3,19x2,2x1)9 = (5x5,4,3,59x2,36x1)9

209 = (19,17,14,13,7x12,11,2x9,5x7,3x5,2x2)9 = (19,17,2x14,13,5x11,7x10,4x5,4,3,1)9

229 = (2x19,3x17,4x15,2x12,11,10,8,2x7,2x6,3x4,3,2x2)9

= (2x19,18,3x16,3x15,14,13,2x12,3x8,7,6,4x5,3,2x1)9

= (2x19,18,17,16,2x15,4x14,13,12,2x9,8,2x6,5,3x4,2x3,1)9

= (2x19,2x18,15,4x14,3x13,12,3x11,4x6,4,2,3x1)9

= (20,3x18,15,14,13,5x12,11,6x9,4x4,3,2)9

- to base x = 25

710 = (27x5,17x4,16x3,32x2,30x1)10

= (2x6,14x5,23x4,11x3,56x2,16x1)10

= (4x6,1x5,29x4,6x3,80x2,2x1)10

810 = (1x7,11x6,8x5,45x4,14x3,1x2,9x1)10 = (3x7,2x6,6x5,44x4,11x3,3x2,20x1)10

910 = (2x8,2x7,9x6,23x5,5x4,5x3,7x2,3x1)10 = (2x8,4x7,21x5,4x4,2x3,9x2,14x1)10

1810 = (16,15,4x14,3x13,5x12,10,9,2x7,7x5,4x4,3x2,2x1)10

= (17,4x14,6x12,10,4x9,8,3x7,6,2x5,2x4,2x3,3x2,4x1)10

2010 = (2x18,17,15,14,13,12,4x9,8,6x7,11x6,5,2x3,2x2)10

= (19,2x17,12,10,3x8,6x7,6,5x5,3x4,3,2,9x1)10

- to base x = 20

1711 = (2x15,2x14,2x13,4x12,7x11,10,7x9,12x4,3x3,2x2,4x1)11

= (3x15,3x13,12,7x11,4x9,9x8,7,3x6,5,2x4,5x3,6x2,1)11

= (16,15,13,8x12,10,4x9,6x8,8x7,2x6,2x5,5x4,2x3,4x2,1)11

2011 = (18,17,4x16,7x14,3x13,2x12,2x11,9,9x8,6x6,4,3x3,2x2)11

= (19,17,3x16,12,4x10,3x9,2x8,2x7,8x6,2x5,3x3,12x2)11

- to base x = 20

512 = (7x4,238x3,53x2,67x1)12 = (9x4,175x3,35x2,146x1)12 = (11x4,112x3,1x2,225x1)12

1112 = (10x9,22x7,4x6,3x5,11x4,3,15x2,13x1)12

= (10,5x9,10x8,7,8x6,32x5,4,2x3,10x2,9x1)12

1212 = (7x10,4x9,8x8,16x7,6x6,8x5,9x4,2x3,4x2,2x1)12

= (2x11,2x10,9,4x8,2x7,24x6,8x5,3x4,14x3,2,5x1)12

1312 = (5x11,5x10,6x9,13x8,8x6,2x5,6x4,18x3,13x2,2x1)12

= (12,3x11,17x9,2x8,12x6,7x5,7x4,20x3,7x2,2x1)12

= (12,3x11,3x10,3x9,16x8,7x6,19x5,4,12x3,2,12x1)12

= (12,3x11,4x10,3x9,7x7,10x6,3x5,4,9x3,29x2,8x1)12

= (2x12,11,2x10,9,19x6,15x5,10x3,6x2,22x1)12

1512 = (14,3x12,14x11,2x10,9,8,7,4x5,9x4,17x3,2,12x1)12

= (2x14,3x11,10,20x9,4x8,9x6,10x4,3,7x2,9x1)12

= (2x14,4x11,2x10,6x9,7x7,5x6,9x5,15x4,3,9x2,6x1)12

• to base x = 20

others :

1014 = (5x8,80x7,300x6,39x5,17x4,15x3,8x2,26x1)14

= (9,8,67x7,347x6,15x5,4,6x3,21x2,31x1)14