There are 31 known primitive solutions to $a^4+b^4+c^4 = d^4$ with $d<10^{28}.$ (Update: As of Feb. 21, there are now 93. See this MSE table.) Old statistics are,
\begin{array}{|c|c|} \hline \text{Range} & \text{# of sol} \\ \hline 10^5-10^7 & \;3 \\ \hline 10^7-10^9 & 12 \\ \hline 10^9-10^{28} & 16 \\ \hline \text{Total} & 31 \\ \hline\end{array}
A brute-force search was done for $d<10^9$, and explains why it is more dense than the range $10^9-10^{28}$ (where solutions were found mostly using several elliptic curves). The objective of this post is to find more $d<10^{28}$ using other elliptic curves.
I. Curve 1
Given $a^4+b^4+c^4 = d^4$ in the form,
$$(15968 - 2334 v - 59v^2)^4 + (7068 + 3082 v + 10v^2)^4 + t^4 = (22628 + 54 v + 159v^2)^4$$
where,
$$4(110301312 + 10244932v - 1285119v^2 + 5299v^3 - 6260v^4) = t^2$$
For any $v$, the terms $(a,b,c,d)$ satisfy the simple relationship,
$$m_1=\frac{(a+b)^2-c^2-d^2}{a^2+ab+b^2+(a+b)d}=-\frac{9}{20}$$
For infinitely many $v$, then $t$ is also rational. "Smallish" solutions are,
$$v = \frac{77 }{9}, \frac{171808 }{16161}, \frac{2465138 }{293763}, \ \frac{5207881}{ 1383765}, \frac{13617482}{ 1280007}, \frac{1251197642}{314528967}$$
$$\color{red}{-v} = \frac{1022}{ 243}, \frac{50191 }{8685}, \frac{128416}{ 29685}, \frac{1116448 }{2565009}, \frac{10267558}{ 1775127}, \frac{237282598}{377952087}\quad$$
For example, let $v = -\frac{1022}{ 243}$ and $v=\frac{77 }{9}$, substituting $(v,t)$ then, after removing common factors, yields the $1$st and $2$nd smallest solutions,
$$95800^4+ 217519^4+ 414560^4= 422481^4\\ 673865^4 + 1390400^4 + 2767624^4 = 2813001^4$$
This curve has been well-explored and the 12 points yield 6 primitive $(a,b,c,d)$ found by Tomita with $d<10^{20}$. It may serve to show how the growth of the $v_k$ seems reasonably slow and it is hoped that the next curves will be similar.
II. Curve 2
$$(11980 - 1673 v - 54v^2)^4 + (-36 + 2321 v - 3v^2)^4 + t^4 = (24677 + 203 v + 71v^2)^4$$
where,
$$591800025 + 20030510v + 1671327v^2 + 92762v^3 - 4112v^4 = t^2$$
The terms satisfy,
$$m_2=\frac{(a+b)^2-c^2-d^2}{a^2+ab+b^2+(a+b)d}=-\frac{29}{12}$$
Only two small solutions are known. Can you find more?
$$v=-\frac{2020}{127}, \frac{76164}{2063}$$
After removing common factors, either $v$ will yield the $3$rd smallest,
$$1705575 ^4 + 5507880^4 + 8332208^4 = 8707481^4$$
III. Curve 3
$$(- 1058960 + 11203324v + 178500100v^2)^4 + (518320 + 16483396v - 294176372v^2)^4 + t^4 = (1304433 - 27003006v + 345712797v^2)^4$$
where,
$$-1251219988511 + 78204922436804v - 1649103906705762v^2 + 19988050672538996v^3 - 76026722992074935v^4=t^2$$
The terms satisfy,
$$m_3=\frac{(a+b)^2-c^2-d^2}{a^2+ab+b^2+(a+b)d}=-\frac{93}{80}$$
Only four solutions are known,
$$v=\frac{12040}{ 110133}, \frac{10691}{353335}, \frac{737109}{ 5187253}, \frac{7880680}{ 207097317}$$
The 4 $v_k$ will give 2 primitive $(a,b,c,d)$ with the first $v$ yielding the $4$th smallest,
$$5870000^4 + 8282543^4 + 11289040^4 = 12197457^4$$
V. Curve 4
$$(- 20150032 + 34614497v + 67829197v^2)^4 + (5444488 + 58502527v - 148896448v^2)^4 + t^4 = (30141789 - 124521519v + 197120781v^2)^4$$
where,
$$-812211871484873 + 7722674874928166v - 27864960882719827v^2 + 48860237516933014v^3 - 31577722089576368v^4 = t^2$$
The terms satisfy,
$$m_4=\frac{(a+b)^2-c^2-d^2}{a^2+ab+b^2+(a+b)d}=-\frac{136}{133}$$
Only four solutions are known,
$$v = \frac{17611}{74168}, \frac{438773}{1417384}, \frac{3258337}{6134331}, \frac{298041047}{414260301}$$
The 4 $v_k$ will also give 2 primitive $(a,b,c,d)$, with the second $v$ yielding the $5$th smallest,
$$4479031^4 + 12552200^4 + 14173720^4 = 16003017^4$$
V. Question
We skip Curve 1. Can more rational points to Curves 2,3,4 be found so it yields primitive solutions to $a^4+b^4+c^4 = d^4$ with the restriction $|d|<10^{28}$ so that it will fit in this table?
P.S. The list has no additions since 2015.
