1.Introduction

I found many parameter solutions for A14 + A24 + A34 - A44 - A54 - A64 +/- A74 = n. (n< 100)

Especially, (6x2-2)4 + (6x2+2)4 - (6x2-1)4 - (6x2+1)4 - (6x)4 = 30 is very cute!

30 is representable by 5 fourth powers in the infinity!

2. Result
          

(648x4-1)4 + (432x4+1)4 + (216x4+2)4 - (2163x4)4 - (432x4-2)4 - (216x4-1)4 - (12x)4 = 1 
 
(6750x4)4 + (3375x4+2)4 + (3375x4+3)4 - (6750x4+1)4 - (3375x4-3)4 - (3375x4)4 - (30x)4 = 15 
  
(24x2+6)4 + (12x2+6)4 + (12x2-3)4 - (24x2+7)4 - (12x2-1)4 - (12x2-4)4 + (12x)4 = 15

(288x4+1)4 + (288x4+2)4 + (192x4+3)4 - (288x4)4 - (192x4)4 - (288x4+3)4 - (96x3)4 = 17 

(6x2-2)4 + (6x2+2)4 - (6x2-1)4 - (6x2+1)4 - (6x)4 = 30

(5464513584x4+15)4 + (2732256792x4+28)4 + (1366128396x4-5)4 - (5464513584x4+19)4 - (2732256792x4-8)4 - (1366128396x4+27)4 - (2796x)4 = 48 

(224563731750x4-13)4 + (187136443125x4)4 + (74854577250x4+14)4 - (224563731750x4-3)4 - (187136443125x4-16)4 - (74854577250x4-6)4 - (6690x)4 = 64 

(107811x4-3)4 + (35937x4+4)4 + (71874x4-1)4 - (107811x4-2)4 - (35937x4+1)4 - (71874x4-4)4 - (66x)4 = 65 

(2025x4-3)4 + (675x4+5)4 + (1350x4+1)4 - (2025x4-1)4 - (675x4-1)4 - (1350x4-5)4 - (30x)4 = 80 

(648x4+3)4 + (216x4+2)4 + (432x4+4)4 - (648x4+4)4 - (216x4-1)4 - (432x4+1)4 - (12x)4 = 95 

(9126x4-3)4 + (4563x4-11)4 + (4563x4+11)4 - (9126x4-5)4 - (4563x4+3)4 - (4563x4+13)4 + (78x)4 = 96

(648x4-2)4 + (216x4-3)4 + (432x4-3)4 - (648x4-3)4 - (216x4)4 - (432x4)4 + (12x)4 = 97 





3. Method

A14 + A24 + A34 - A44 - A54 - A64 +/- A74 = n......(1)

(a1x+c1)4+(a2x+c2)4+(a3x+c3)4-(a1x+c4)4-(a2x+c5)4-(a3x+c6)4

=(-4c4a13-4c5a23+4c1a13+4c2a23+4c3a33-4c6a33)x3
+(-6c42a12-6c52a22-6c62a32+6c12a12+6c22a22+6c32a32)x2
+(4c13a1-4c43a1-4c53a2-4c63a3+4c23a2+4c33a3)x
+c14+c24+c34-c44-c54-c64..........................(2)

1. Case 1
   Decide (a1,a2,a3) and (c1,c2,c3,c4,c5,c6) to make the coefficient of x3 and x2 to 0.
   Set n=c14+c24-c44-c54-c64+c34.
   We can obtain a parameter solution of (1).
   n=1,15,48,64,65,80,95,96,97    
 
2. Case 2
   Decide (a1,a2,a3) and (c1,c2,c3,c4,c5,c6) to make the coefficient of x3 and x to 0.
   Set n=c14+c24-c44-c54-c64+c34.
   We can obtain a parameter solution of (1).
   n=15 and 30

3. Case 3
   Decide (a1,a2,a3) and (c1,c2,c3,c4,c5,c6) to make the coefficient of x2 and x to 0.
   Set n=c14+c24-c44-c54-c64+c34.
   We can obtain a parameter solution of (1).
   n=17

4. Consideration of c14+c24+c34-c44-c54-c64 mod 16

   Generally, n^4=0,1 mod 16.
   So, c14+c24+c34-c44-c54-c64 mod 16 = 0,+/-1,+/-2,+/-3.
   At n < 100, n=4, 5, 6, 7, 8, 9, 10, 11, 12, 20, 21, 22, 23, 24, 25, 26, 27, 28, 36, 37, 38,
   39, 40, 41, 42, 43, 44, 52, 53, 54, 55, 56, 57, 58, 59, 60, 68, 69, 70, 71, 72, 73, 74, 75,
   76, 84, 85, 86, 87, 88, 89, 90, 91, 92 are not representable by this method.

   About other number n=2,3,13,14,18,19,29,31,32,33,34,35,45,46,47,49,50,51,61,62,63,66,67,
   77,78,79,82,83,93,94,98,99,I dont know that n are representable or not by this method.

4. Example

 (648x4-1)4 + (432x4+1)4 + (216x4+2)4 - (2163x4)4 - (432x4-2)4 - (216x4-1)4 - (12x)4 = 1

  1<=x<=20

    x       
 
    1       6474 +       4334 +       2184 -       6484 -       4304 -       2154 -  124 = 1
    2     103674 +      69134 +      34584 -     103684 -      69104 -      34554 -  244 = 1
    3     524874 +     349934 +     174984 -     524884 -     349904 -     174954 -  364 = 1
    4    1658874 +    1105934 +     552984 -    1658884 -    1105904 -     552954 -  484 = 1
    5    4049994 +    2700014 +    1350024 -    4050004 -    2699984 -    1349994 -  604 = 1
    6    8398074 +    5598734 +    2799384 -    8398084 -    5598704 -    2799354 -  724 = 1
    7   15558474 +   10372334 +    5186184 -   15558484 -   10372304 -    5186154 -  844 = 1
    8   26542074 +   17694734 +    8847384 -   26542084 -   17694704 -    8847354 -  964 = 1
    9   42515274 +   28343534 +   14171784 -   42515284 -   28343504 -   14171754 - 1084 = 1
   10   64799994 +   43200014 +   21600024 -   64800004 -   43199984 -   21599994 - 1204 = 1
   11   94873674 +   63249134 +   31624584 -   94873684 -   63249104 -   31624554 - 1324 = 1
   12  134369274 +   89579534 +   44789784 -  134369284 -   89579504 -   44789754 - 1444 = 1
   13  185075274 +  123383534 +   61691784 -  185075284 -  123383504 -   61691754 - 1564 = 1
   14  248935674 +  165957134 +   82978584 -  248935684 -  165957104 -   82978554 - 1684 = 1
   15  328049994 +  218700014 +  109350024 -  328050004 -  218699984 -  109349994 - 1804 = 1
   16  424673274 +  283115534 +  141557784 -  424673284 -  283115504 -  141557754 - 1924 = 1
   17  541216074 +  360810734 +  180405384 -  541216084 -  360810704 -  180405354 - 2044 = 1
   18  680244474 +  453496334 +  226748184 -  680244484 -  453496304 -  226748154 - 2164 = 1
   19  844480074 +  562986734 +  281493384 -  844480084 -  562986704 -  281493354 - 2284 = 1
   20 1036799994 +  691200014 +  345600024 - 1036800004 -  691199984 -  345599994 - 2404 = 1