Diophantine Equations (4th Powers)

"On the Clustering of Sums of Quartic Sextuples"

by Titus Piezas III

This is a short side article to the main paper "Ramanujan and the Quartic Equation 2^4+2^4+3^4+4^4+4^4=5^4". Originally, this was to be a section there but because of space constraints was placed here.

An observation made by the author in the course of studying quartic sextuples was that there seem to be a preference for certain sums. A brute-force, exhaustive search was done for (a,b,c,d,e,f) as primitive solutions to a^4+b^4+c^4+d^4+e^4=f^4. For f<146, we have the first 43 solutions,

b
2
2
6
10
21
2
8
4
13
14
4
6
8
8
39
4
16
13
42
24
12
16
35
48
52
15
14
28
28
36
31
38
11
42
12
24
23
32
48
17
19
96
46

c
3
6
8
10
22
12
24
26
16
28
32
31
13
12
44
32
24
32
48
38
54
22
52
66
57
50
26
63
63
56
48
58
26
44
28
52
46
53
64
68
48
96
62

d
4
12
9
17
26
24
36
27
44
33
34
44
28
32
46
32
36
34
51
68
54
38
60
67
74
50
54
82
72
84
58
71
84
84
78
74
52
116
87
104
74
108
63

e
4
13
14
30
28
44
38
42
48
52
51
46
54
64
52
63
63
84
78
73
77
84
80
78
76
100
103
88
94
88
112
108
118
117
131
131
144
126
138
132
142
108
142

f
5
15
15
31
35
45
45
45
55
55
55
55
55
65
65
65
65
85
85
85
85
85
89
95
95
103
105
105
105
105
115
115
125
125
135
135
145
145
145
145
145
145
145

a
2
2
4
10
4
1
1
4
2
2
4
4
4
1
2
4
12
2
4
8
12
13
10
6
22
4
6
6
22
23
4
14
8
18
4
12
2
4
6
8
14
17
26

Table 1

Why is there a clustering? Or why do some sums appear again and again? The odds that the sum of five 4th powers of random integers below a certain range will be itself a 4th power may not be so high. Yet some 4th powers seem to have a tendency to be decomposable into 4th powers in much more than one way! Compare to the first 15 smallest primitive solutions in positive integers to cubic quintuples a^3+b^3+c^3+d^3=e^3:

e
7
12
13
13
14
18
18
20
23
24
25
26
26
28
28

d
6
11
12
10
13
14
17
14
18
19
24
21
21
26
27

a
1
3
1
5
2
1
4
11
6
5
7
3
5
1
2

b
1
3
5
7
3
7
7
12
14
14
9
15
11
10
4

c
5
7
7
9
8
14
8
13
15
16
9
17
19
15
13

Table 2

Of course, if we extend Table 2 also to the first 43 solutions, there is also some sort of "clustering" but not as dense. And since one can see the average "spacing" is quite near anyway, for the same sums to appear might not be so unusual. For the quartic sums of Table 1, the spacing is farther yet the clustering is denser. Why? Other than multiple representations as p^2+3q^2 pointed out in the main paper and which applies only to certain sextuples, there might be another algebraic reason for it. It might be productive to extend Table 1 to f < 1000 to see if this phenomenon continues.