1.Introduction

We extended the previous result X^3 + Y^3 + Z^3 + W^3 = 1,
and show that X^3 + Y^3 + Z^3 + W^3 = 3n^2+3n+1 has infinitely many integral solutions.
     
     
2.Theorem
     
Diophantine equation X^3 + Y^3 + Z^3 + W^3 = 3n^2+3n+1 has infinitely many integral solutions.

X = -2n^2-3n-1+2a^2
Y = -2n^2-n+2a^2
Z =  2n^2+2n+1-2a^2+a
W =  2n^2+2n+1-2a^2-a

a,n is arbitrary.
 
Proof.

Let X = p+2a^2, Y = q+2a^2, Z = r-2a^2+a, W = r-2a^2-a...................................(1)
We obtain
X^3 + Y^3 + Z^3 + W^3 = (12p+24r-12+12q)a^4+(6q^2+6p^2-12r^2+6r)a^2+2r^3+p^3+q^3.........(2)

From (12p+12q+24r-12)=0, r = -1/2p-1/2q+1/2.

Substitute r to (6q^2+6p^2-12r^2+6r)=0, then p^2+p-2pq+q+q^2 = 0.........................(3)

Solution of equation (3) is given (p,q)=( -(n+1)(2n+1), -n(2n+1) ).

Then q^3+2r^3+p^3 becomes to 3n^2+3n+1.

Thus we obtain a parametric solution below.

X = -2n^2-3n-1+2a^2
Y = -2n^2-n+2a^2
Z =  2n^2+2n+1-2a^2+a
W =  2n^2+2n+1-2a^2-a

Hence X^3 + Y^3 + Z^3 + W^3 = 3n^2+3n+1 has infinitely many integral solutions.

Q.E.D.



3.Examples

3n^2+3n+1<100

(-1+2a^2)^3 + (2*a^2)^3 + (1-2a^2+a)^3 + (1-2a^2-a)^3 = 1

(-6+2a^2)^3 + (-3+2a^2)^3 + (5-2a^2+a)^3 + (5-2a^2-a)^3 = 7

(-15+2a^2)^3 + (-10+2a^2)^3 + (13-2a^2+a)^3 + (13-2a^2-a)^3 = 19

(-28+2a^2)^3 + (-21+2a^2)^3 + (25-2a^2+a)^3 + (25-2a^2-a)^3 = 37

(-45+2a^2)^3 + (-36+2a^2)^3 + (41-2a^2+a)^3 + (41-2a^2-a)^3 = 61

(-66+2a^2)^3 + (-55+2a^2)^3 + (61-2a^2+a)^3 + (61-2a^2-a)^3 = 91








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