1.Introduction

Stephane Vandemergel gave a parametric solution of {a^4 + b^4 + c^4 = d^4 + e^4 + f^4, abc=def} as follows.
{a,b,c}={rp, sp, q^2},{d,e,f}={rq, sq, p^2} where p^4+q^4=r^4+s^4.(Guy's book[1])

We show a parametric solution of {a^4 + b^4 + c^4 + d^4 = e^4 + f^4 + g^4 + h^4, abcd=efgh}.


2.Theorem
        
     

    {a^4 + b^4 + c^4 + d^4 = e^4 + f^4 + g^4 + h^4, abcd=efgh} has a follwing parametric solution.


    a = p(m + r^4 - q^4)          e = p(m - r^4 + q^4)
    b = q(m + p^4 - w^4)          f =-q(-m + p^4 - w^4)
    c =-r(-m + p^4 - w^4)         g = r(m + p^4 - w^4)
    d = w(m - r^4 + q^4)          h = w(m + r^4 - q^4)
    
    Condition: p^4 + q^ 4 = r^4 + w^4

 
Proof.

a^4 + b^4 + c^4 + d^4 = e^4 + f^4 + g^4 + h^4.................................(1)
abcd=efgh.....................................................................(2)
a=pm+s, b=qm+t, c=rm+u, d=wm+v, e=pm-s, f=qm-t, g=rm-u, h=wm-v................(3)

Substitute (3) to (2), and simplifying (2), we obtain

2m(m^2pquw+m^2pqrv+m^2ptrw+m^2*sqrw+ptuv+squv+stuw+strv)=0....................(4)

Equating to zero (4), then we obtain

t= -qu/r, s= -pv/w

Substitute s and t to (1), we obtain

-8(q^4w^3r^2u-w^6r^3v+p^4r^3w^2v-w^3r^6u)m^3
-8(q^4w^3u^3-w^4r^3v^3+p^4r^3v^3-w^3r^4u^3)m=0................................(5)

Equating to zero the coefficient of m^3 in (5), then we obtain

u = -r(-w^4+p^4), v = w(q^4-r^4)
Finally, (1) becomes to as follows,

8m(q-r)(q+r)(q^2+r^2)(p-w)(p+w)(p^2+w^2)(q^4+p^4-w^4-r^4)(-q^4+p^4-w^4+r^4)=0

Hence, if p^4+q^4=r^4+w^4 then (1) becomes to zero and obtain a parametric solution.


   
Q.E.D.　
 
　　                  
       
3.Example


Case 1: We use Euler's solution for p^4+q^4=r^4+w^4.
p = m^7+m^5-2m^3+3m^2+m
q = m^6-3m^5-2m^4+m^2+1
w = m^7+m^5-2m^3-3m^2+m
r = m^6+3m^5-2m^4+m^2+1

a = 24m^29+96m^27-48m^25+72m^24-384m^23+216m^22+312m^21-216m^20+888m^19-576m^18-672m^17+864m^16
   -552m^15+864m^14+672m^13-576m^12+48m^11-216m^10-192m^9+216m^8+25m^7+72m^6+25m^5-2m^3+3m^2+m
   
b = 24m^28-72m^27+24m^26-216m^25-192m^24+216m^23+48m^22+576m^21+672m^20-864m^19-552m^18-864m^17
   -672m^16+576m^15+888m^14+216m^13+312m^12-216m^11-384m^10-72m^9-48m^8+97m^6-3m^5+22m^4+m^2+1
   
c =-24m^28-72m^27-24m^26-216m^25+192m^24+216m^23-48m^22+576m^21-672m^20-864m^19+552m^18-864m^17
   +672m^16+576m^15-888m^14+216m^13-312m^12-216m^11+384m^10-72m^9+48m^8-95m^6+3m^5-26m^4+m^2+1
   
d =-24m^29-96m^27+48m^25+72m^24+384m^23+216m^22-312m^21-216m^20-888m^19-576m^18+672m^17+864m^16
   +552m^15+864m^14-672m^13-576m^12-48m^11-216m^10+192m^9+216m^8-23m^7+72m^6-23m^5-2m^3-3m^2+m
   
e =-24m^29-96m^27+48m^25-72m^24+384m^23-216m^22-312m^21+216m^20-888m^19+576m^18+672m^17-864m^16
   +552m^15-864m^14-672m^13+576m^12-48m^11+216m^10+192m^9-216m^8-23m^7-72m^6-23m^5-2m^3+3m^2+m
   
f =-24m^28+72m^27-24m^26+216m^25+192m^24-216m^23-48m^22-576m^21-672m^20+864m^19+552m^18+864m^17
   +672m^16-576m^15-888m^14-216m^13-312m^12+216m^11+384m^10+72m^9+48m^8-95m^6-3m^5-26m^4+m^2+1
   
g = 24m^28+72m^27+24m^26+216m^25-192m^24-216m^23+48m^22-576m^21+672m^20+864m^19-552m^18+864m^17
   -672m^16-576m^15+888m^14-216m^13+312m^12+216m^11-384m^10+72m^9-48m^8+97m^6+3m^5+22m^4+m^2+1
   
h = 24m^29+96m^27-48m^25-72m^24-384m^23-216m^22+312m^21+216m^20+888m^19+576m^18-672m^17-864m^16
   -552m^15-864m^14+672m^13+576m^12+48m^11+216m^10-192m^9-216m^8+25m^7-72m^6+25m^5-2m^3-3m^2+m

 




4.Reference

[1] Richard K. Guy: Unsolved Problems in Number Theory