Summary of Euler lengths for small powers
For comments on the following table, see below.
x \ m= 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
4 
5 
16 
19 
15 
66 
121 
146 
199 
277 
911 
5 
7 
5 
18 
29 
39 
80 
32 
188 
137 
365 
6 
3 
6 
14 
22 
28 
55 
102 
144 
145 
456 
7 
4 
6 
12 
21 
26 
40 
67 
122 
51 
196 
8 
5 
16 
9 
15 
21 
44 
63 
89 
88 
157 
9 
3 
6 
11 
19 
20 
34 
46 
56 
70 
118 
10 
5 
5 
9 
15 
18 
40 
32 
56 
85 
66 
11 
5 
7 
13 
17 
22 
40 
37 
44 
75 
79 
12 
3 
6 
6 
15 
22 
34 
42 
34 
61 
66 
13 
4 
6 
12 
16 
22 
38 
33 
46 
52 
78 
14 
4 
6 
8 
14 
20 
36 
34 
39 
38 
66 
15 
5 
5 
10 
15 
21 
36 
29 
23 
46 
66 
16 
5 
16 
9 
13 
14 
34 
29 
34 
40 
54 
17 
5 
6 
8 
12 
17 
17 
29 
34 
46 
53 
18 
3 
6 
10 
11 
14 
34 
27 
34 
39 
44 
19 
3 
6 
10 
11 
14 
29 
26 
33 
38 
42 
20 
3 
5 
9 
13 
17 
34 
24 
34 
42 
46 
21 
4 
6 
10 
14 
16 
27 
16 
23 
39 
42 
22 
5 
7 
10 
12 
17 
33 
25 
33 
35 
44 
23 
4 
6 
7 
13 
14 
29 
19 
32 
36 
40 
24 
3 
6 
6 
13 
16 
33 
19 
34 
35 

25 
3 
5 
9 
12 
13 
32 
20 
26 
31 

26 
4 
6 
7 
12 
15 

15 



27 
3 
6 
9 
12 
13 

18 



28 
4 
6 
8 
12 
12 

21 



29 
3 
6 
9 
12 
15 

21 



30 
3 
5 
6 
11 
13 

17 



31 
4 
5 
9 
11 
15 

19 



32 
4 
16 
6 
11 
13 

18 



33 
4 
6 
8 
11 
14 

19 



34 
4 
6 
7 
11 
13 

19 



35 
4 
5 
7 
13 
14 

18 



36 
3 
6 
6 
11 
13 

17 



37 
4 
6 
8 
11 
13 

17 



38 
3 
6 
7 
11 
14 

18 



39 
4 
6 
8 
10 
12 

19 



40 
3 
5 
7 
11 
12 

18 



41 
3 
6 
7 
11 






42 
3 
6 
7 
11 






43 
4 
6 
9 
11 






44 
3 
7 
8 
11 






45 
3 
5 
9 
10 






The above table highlights a variety of points.
The values for m = 3 are consistent with the conjecture that G(3) £ 4 (see Chapter 7). This is slightly misleading as the above table gives the shortest representation of x^{m} in terms of lesser mth powers, whereas with g(m) and G(m), x^{m} is itself the shortest.
For m = 4, the values indicate that when x is a power of 2, at least 16 powers are required, consistent with Davenport’s result that G(4) = 16, and otherwise, that at most 6 are needed in the long term. The trends for higher powers are similar in that they all quickly settle around a plateau of values. One wonders how close these values come to suggesting G(m) ?
The table also suggests that m = 9 in an inherently "better" power for obtaining Euler representations than m = 8, with solutions being found quicker and of shorter length on average for a particular base. However, as seen in Chapter 1, the m = 8 case allows additional restrictive conditions to be applied when searching for the shortest possible.
On a more speculative note, there are vague patterns and instances that suggest the following:
For each m,
are there "favoured" values for Euler lengths, that recur regularly ?
is there any significance in common Euler lengths for consecutive bases ?
is there any significance in exceptional values such as that for 17^{8} ?