Is the quartic diophantine equation $a^4+nb^4 = c^4+nd^4$ solvable for any integer $n$?

Solution 1:

I used a Maple program to first find the set $S$ of all positive integers $\le 10^{12}$ that can be written as the difference of two fourth powers (there are $649913$ of them), then for successive integers $n$ find (if it exists) the least member of $S \cap (S/n$). Thus for $n=2$ the result was $179727600$, corresponding to $179727600 = 116^4 - 34^4 = (139^4 - 61^4)/2$. Hmm, this sequence belongs in the OEIS (it doesn't seem to be there yet).

There were $133$ cases up to $n=1000$ where $S \cap (nS)$ was empty, of which the first was $n=206$. That doesn't say there is no solution for $n=206$, just that any solution will have $c^4 - a^4 > 10^{12}$.

Solution 2:

According to this paper (and https://arxiv.org/pdf/1701.02602.pdf),
the integer solutions of this equation (for that time) are known for all positive integer values of $n \le 5000$ and $a,b,c,d \le 100\:000$, except for some numbers.

These exceptions $n$ are listed here:

$1198, 1787, 1987;$

$2459, 2572, 2711, 2797, 2971;$

$3086, 3193, 3307, 3319, 3334, 3347, 3571, 3622, 3623, 3628, 3644, 3646, 3742, 3814, 3818, 3851, 3868, 3907, 3943, 3980;$

$4003, 4006, 4007, 4051, 4054, 4099, 4231, 4252, 4358, 4406, 4414, 4418, 4478, 4519, 4574, 4583, 4630, 4643, 4684, 4870, 4955, 4999.$

(Update)

Great database of solution for this equation by Seiji Tomita contains solutions for all $n< 20000$. Thanks, Sam (follow link of his answer).

We'll use $2$ similar values (to estimate "size" of the solution): $$ \mu_{\square} = \max\{a,b,c,d\};\tag{1}$$ $$ \mu_{\bigcirc} = \sqrt[4]{a^4+nb^4}.\tag{2}$$

Value $(1)$ was used mostly in current research; value $(2)$ was used in Seiji Tomita research.

Current results:

• the smallest (by $\mu_{\square}$ and $\mu_{\bigcirc}$) solutions were found for listed above values of $n$;
• for each $n\le 5000$ there exists at least one solution with $\mu_{\square}<620\:000$ and $\mu_{\bigcirc}<1\:100\:000$.

The table below contains the smallest solutions for listed values. And small improvement for value $n=967$.

\begin{array}{|r|l|l|l|l|} \hline n & (a,b,c,d) & \mu_{\bigcirc} & \mbox{by ...} & \mbox{notes} \\ \hline 967 & (251477, 18748, 146927, 44086) & \approx 253334 & & new \\ \hline 1198 & (177233, 32517, 134247, 35951) & \approx 219611 & & new \\ \hline 1787 & (110571, 9891, 78851, 16357) & \approx 113607 & & JW \\ \hline 1987 & (182489, 867, 4289, 27333) & \approx 182489 & & JW \\ \hline 2459 & (213308, 2587, 194886, 22479) & \approx 213311 & & JW \\ \hline 2572 & (308817, 54121, 271169, 56253) & \approx 420150 &\mu_{\square} & new \\ & (394714, 20667, 286866, 51413) & \approx 396608 &\mu_{\bigcirc}& new \\ \hline 2711 & (270983, 20284, 200731, 35338) & \approx 276573 & & JW \\ \hline 2797 & (159689, 6263, 159577, 6841) & \approx 159953 & & new \\ \hline 2971 & (109288, 18485, 92740, 19339) & \approx 148746 & & JW \\ \hline 3086 & (376829, 16357, 265733, 47263) & \approx 377857 & & ST + \\ \hline 3193 & (102997, 19519, 10091, 20609) & \approx 154920 &\mu_{\square} & maybe\;new \\ & (125429, 11042, 437, 17434) & \approx 131053 &\mu_{\bigcirc} & ST + \\ \hline 3307 & (337537, 27465, 251109, 42595) & \approx 349156 & & JW \\ \hline 3319 & (105539, 5039, 53773, 13727) & \approx 105991 & & JW \\ \hline 3334 & (202509, 8212, 189173, 18798) & \approx 202964 & & ST + \\ \hline 3347 & (225431, 2704, 158491, 27634) & \approx 225435 & & JW \\ \hline 3571 & (250803, 20129, 213427, 28833) & \approx 259617 & & JW \\ \hline 3622 & (565013, 110014, 347731, 114284)& \approx 891789 &\mu_{\square}& new \\ & (843607, 64857, 217639, 111921) & \approx 869113 &\mu_{\bigcirc}& new \\ \hline 3623 & (138251, 29344, 71883, 30228) & \approx 235034 & & JW \\ \hline 3628 & (617824, 57931, 106276, 84667) & \approx 657212 & & ST + \\ \hline 3644 & (118519, 9823, 43639, 15809) & \approx 123315 & & new \\ \hline 3646 & (105497, 7993, 95033, 11191) & \approx 108532 & & new \\ \hline 3742 & (322387, 172226, 250139, 172316)& \approx 1.34813\times10^6 &\mu_{\square}& new \\ & (1021473, 98254, 187193, 139968)& \approx 1.09496\times10^6 &\mu_{\bigcirc} & ST + \\ \hline 3814 & (96317, 7481, 6661, 12661) & \approx 99498.2 & & new, (small?) \\ \hline 3818 & (67219, 3637, 39021, 8373) & \approx 67762.3 & & new, (small?) \\ \hline 3851 & (108451, 2221, 22483, 13763) & \approx 108469 & & JW \\ \hline 3868 & (141656, 4633, 137788, 10327) & \approx 141812 & & new \\ \hline 3907 & (154274, 8879, 83948, 19291) & \approx 155901 & & JW \\ \hline 3943 & (109819, 20074, 54617, 21068) & \approx 167424 & & JW \\ \hline 3980 & (93077, 10904, 66411, 12948) & \approx 107048 & & ST +, (small?) \\ \hline 4003 & (156337, 137593, 115867, 137603)& \approx 1.09456\times10^6 &\mu_{\square} & new \\ & (182563, 5887, 137677, 20849) & \approx 182760 &\mu_{\bigcirc} & JW \\ \hline 4006 & (142183, 67157, 22063, 67241) & \approx 534949 &\mu_{\square} & maybe\;new \\ & (143746, 343, 71638, 17783) & \approx 143746 &\mu_{\bigcirc} & ST + \\ \hline 4007 & (148131, 9592, 124089, 16234) & \approx 150674 & & JW \\ \hline 4051 & (368461, 137025, 11973, 137465) & \approx 1.09669\times10^6 &\mu_{\square} & new \\ & (442912, 213, 410504, 39723) & \approx 442912 &\mu_{\bigcirc} & JW \\ \hline 4054 & (406147, 37967, 239933, 53137) & \approx 434477 & & new \\ \hline 4099 & (108466, 8549, 63692, 13687) & \approx 112522 & & JW + \\ \hline 4231 & (165485, 1073, 164533, 7975) & \approx 165485 & & JW + \\ \hline 4252 & (422378, 63381, 350094, 66949) & \approx 562965 & & new \\ \hline 4358 & (100212, 95919, 91496, 95921) & \approx 779392 &\mu_{\square} & new \\ & (113251, 10894, 30563, 15074) & \approx 122594 &\mu_{\bigcirc} & new \\ \hline 4406 & (355473, 45564, 60271, 53068) & \approx 432399 & & new \\ \hline 4414 & (110846, 6721, 92198, 11873) & \approx 112464 & & new \\ \hline 4418 & (337601, 62681, 112753, 65441) & \approx 533793 &\mu_{\square} & new \\ & (399547, 17732, 184381, 48658) & \approx 401248 &\mu_{\bigcirc} & new \\ \hline 4478 & (303719, 17780, 249983, 32590) & \approx 307636 & & new \\ \hline 4519 & (155665, 20255, 16057, 23369) & \approx 191605 &\mu_{\square} & JW \\ & (166779, 571, 112551, 19193) & \approx 166779 &\mu_{\bigcirc} & JW +\\ \hline 4574 & (218107, 10033, 56333, 26627) & \approx 219215 & & new \\ \hline 4583 & (232822, 7771, 216312, 20217) & \approx 233152 & & JW \\ \hline 4630 & (489443, 83505, 63483, 88381) & \approx 729055 & & new \\ \hline 4643 & (101129, 9634, 73271, 12566) & \approx 109657 & & JW \\ \hline 4684 & (166861, 14052, 111837, 20338) & \approx 175923 & & ST + \\ \hline 4870 & (524118, 46841, 257242, 66375) & \approx 560794 & & ST + \\ \hline 4955 & (246607, 31000, 36819, 35946) & \approx 301603 & & new \\ \hline 4999 & (131429, 4656, 81439, 15054) & \approx 131687 & & JW + \\ \hline \end{array}

If column "by..." is empty, then solution is smallest by both: $\mu_{\square}$ and $\mu_{\bigcirc}$.

Column "notes":
$ST$: found by Seiji Tomita (according to this link);
$JW$: found by Jaroslaw Wroblewski (according to this link);
$new$: new solution, a.f.a.i.k.;
$+$: confirm that result of Seiji Tomita database is the smallest (by $\mu_{\bigcirc}$), since doubt signs "???" are there;
$(small?)$: solution has $\max\{a,b,c,d\}<100\:000$.