1.Introduction


                n=+/-x1k +/-x2k +/-x3k...+/-xsk

We use v(k) to denote the least value of s for which every n is representable in this manner.

In the case k=3,Demjanenko proved that every integer n is a sum of four cubes
exclude 9m+4 and 9m-4.([1].Demjanenko)

In the case k=4,it seemed that 9<=v(4)<=10.([2]. Hardy)

On the condition of v(4)<=10,I found some polinomial identities below.

For example

(3x+73)4+(2x+53)4-(3x+75)4-(2x+46)4+(x+31)4+(x+22)4-(x+32)4-(x+23)4=48x+1

This identities means that every integer of 48x+1 is representable by eight 4th powers.

734+534-754-464+314+224-324-234=1
764+554-784-484+324+234-334-244=49


I couldn't find the cases of 48x+6,48x+7,48x+8,48x+9,48x+10,48x+22,48x+23,48x+24,
48x-6,48x-7,48x-8,48x-9,48x-10,48x-22,48x-23.

  [1]. Demjanenko：http://www.math.u-bordeaux1.fr/~cohen/sum4cub.ps
　[2]. Hardy and Wright: AN INTRODUCTION TO THE THEORY OF NUMBERS　                  
       
2.Search results




(x+2)4+2(x-1)4-2(x+1)4-(x-2)4=48x
   
(x+2)4+3x4-3(x+1)4-(x-1)4=24x+12

(3x+73)4+(2x+53)4-(3x+75)4-(2x+46)4+(x+31)4+(x+22)4-(x+32)4-(x+23)4=48x+1

(3x-75)4+(2x-46)4-(3x-73)4-(2x-53)4+(x-23)4+(x-32)4-(x-22)4-(x-31)4=48x-1

(6x-139)4+(4x-95)4-(6x-140)4-(4x-92)4+(3x-66)4+(x-17)4-(3x-65)4-(x-20)4=48x+2

(6x+58)4+(4x+40)4-(6x+57)4-(4x+41)4+(3x+25)4+(x+14)4-(3x+31)4-(x+4)4=48x-2

-(2x+3)4-3(x-2)4+(2x+2)4+2(x+3)4+2(x-5)4-(x-6)4=24x+3

-(2x+58)4-(x+35)4+(2x+57)4+(x+34)4-(x+29)4-(x+27)4+2(x+32)4-(x+25)4+(x+26)4=24x-3

-(3x+36)4-(2x+26)4+(3x+37)4+(2x+22)4-(x+16)4-(x+10)4+2(x+15)4-(x+8)4+(x+9)4=24x+4

(3x-36)4+(2x-26)4-(3x-37)4-(2x-22)4+(x-8)4+(x-10)4-(x-9)4-2(x-15)4+(x-16)4=24x-4

-(x+2)4-(-2x+2)4+(x+5)4+(-2x-1)4-81x4-(-3x-6)4+(-3x+1)4+2(-3x-7)4-(-3x-8)4=24x+5

-(x-1)4-(x-3)4+(x+4)4+x4-(-2x+1)4-(-3x-3)4+(-2x+2)4+2(-3x-4)4-(-3x-5)4=24x-5

(3x-12)4+(2x-6)4-(3x-11)4-(2x-9)4+2(x-4)4-(x-5)4-2(x-7)4+(x-8)4=24x+11

-(3x+6)4-(2x+6)4+(3x+7)4+(2x+3)4-2(x+4)4+(x+5)4+2(x-1)4-(x-2)4=24x-11

-(6x+9)4-(4x+8)4+(6x+10)4+(4x+5)4-2(2x+5)4+(2x+6)4+2(2x)4-(2x-1)4=48x+13

(6x-15)4+(4x-8)4-(6x-14)4-(4x-11)4+2(2x-5)4-(2x-6)4-2(2x-8)4+(2x-9)4=48x-13

(6x-30)4+(4x-22)4-(6x-31)4-(4x-19)4+(3x-14)4+(x-1)4-(3x-13)4-(x-4)4=48x+14

(6x+31)4+(4x+19)4-(6x+30)4-(4x+22)4+(3x+13)4+(x+4)4-(3x+14)4-(x+1)4=48x-14

(2x+18)4+(x+4)4-(2x+17)4-(x+3)4+(x+7)4+(x+5)4-(x+11)4-(x+10)4=48x+15

(3x+17)4+(2x+9)4-(3x+16)4-(2x+12)4+(x+4)4+(x+2)4-(x+8)4-(x+1)4=48x-15

(5x+7)4+(4x+9)4-(5x+9)4-(4x+5)4+(3x+7)4+(2x+7)4-(3x+9)4-(2x+1)4=48x+16

(3x+36)4+(3x+26)4-(3x+37)4-(3x+28)4+(2x+25)4+(x+15)4-(2x+15)4-(x+14)4=48x-16

(4x+9)4+(2x+11)4-(4x+10)4-(2x+3)4+(3x+14)4+(3x+7)4-(3x+15)4-(3x+6)4=48x+17

(4x+20)4+(3x+15)4-(4x+21)4-(3x+13)4+(3x+9)4+(2x+12)4-(3x+11)4-(2x+4)4=48x-17

(4x-303)4+(3x-224)4-(4x-302)4-(3x-229)4+(2x-155)4+(x-65)4-(2x-146)4-(x-66)4=48x+18

2(2x-1)4-2x4-(2x)4-(2x-2)4+(x+2)4+(x-2)4=48x+18

(2x+1952)4+(2x+1946)4-(2x+1955)4-(2x+1945)4+(x+993)4+(x+992)4-(x+997)4-(x+972)4=48x-18

(2x+2)4+2x4+(2x)4-2(2x+1)4-(x+2)4-(x-2)4=48x-18