1.Introduction


We show that X^3 + Y^3 + Z^3 + W^3 = 24n^2+2 has infinitely many integral solutions.
     
     
2.Theorem
     
Diophantine equation X^3 + Y^3 + Z^3 + W^3 = 24n^2+2 has infinitely many integral solutions.

X =  4a^2-4n^2+2n
Y =  4a^2-4n^2-2n
Z = -4a^2+2a+4n^2+1
W = -4a^2-2a+4n^2+1

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(2n-1), -n(2n+1) ).

Then q^3+2r^3+p^3 becomes to 3n^2+1/4.

Thus we obtain a parametric solution below.

X =  4a^2-4n^2+2n
Y =  4a^2-4n^2-2n
Z = -4a^2+2a+4n^2+1
W = -4a^2-2a+4n^2+1

Thus X^3 + Y^3 + Z^3 + W^3 = 24n^2+2 has infinitely many integral solutions.

Q.E.D.


3.Examples

24n^2+2<100

(4a^2-2)^3+(4a^2-6)^3+(-4a^2+2a+5)^3+(-4a^2-2a+5)^3 = 26

(4a^2-12)^3+(4a^2-20)^3+(-4a^2+2a+17)^3+(-4a^2-2a+17)^3 = 98






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