1.Introduction


We give a parametric solution of ax^4 + by^4 + cz^4 = w^2 where a+b+c=r^2.


2.Theorem
        
     

    ax^4 + by^4 + cz^4 = w^2 has a follwing parametric solution.

    x = (-3ar^6-4a^4+4a^3r^2+3a^2r^4)p^4+(12br^4a+16a^3c-16a^3b-12cr^4a)p^3
      +(-12ac^2r^2+12a^2br^2+12a^2cr^2-6br^4a+48a^2cb-24a^2b^2-24a^2c^2-12ab^2r^2-6cr^4a+24cbar^2)p^2
      +(-24c^2br^2+48acb^2-8b^3r^2-48ac^2b+24cb^2r^2+4cr^6-24ac^2r^2+16ac^3+8c^3r^2-16ab^3-4br^6+24ab^2r^2+12b^2r^4-12c^2r^4)p
      +16c^3b+br^6+cr^6+16cb^3+12b^3r^2-24c^2b^2+12c^3r^2-12cb^2r^2-18cr^4b-9c^2r^4-9b^2r^4-4b^4-12c^2br^2-4c^4    
      
    y = (ar^6-4a^4+12a^3r^2-9a^2r^4)p^4+(24a^2br^2-4ar^6-16a^3b-24a^2cr^2+12a^2r^4+16a^3c-8a^3r^2)p^3
      +(-12a^2br^2-24cbar^2-24a^2b^2-30cr^4a-24a^2c^2+12ab^2r^2+36a^2cr^2-6br^4a+12ac^2r^2+48a^2cb)p^2
      +(16ac^3+48acb^2+48cbar^2-48ac^2b+12br^4a-16ab^3-48ac^2r^2+12cr^4a)p
      +16c^3b-3br^6+5cr^6+16cb^3+4b^3r^2-24c^2b^2+20c^3r^2+12cb^2r^2-18cr^4b-21c^2r^4+3b^2r^4-4c^4-36c^2br^2-4b^4    

    z = (-ar^6+4a^4-12a^3r^2+9a^2r^4)p^4+(-24a^2br^2-4ar^6+16a^3b+24a^2cr^2+12a^2r^4-16a^3c-8a^3r^2)p^3
      +(24cbar^2+24a^2b^2-36a^2br^2+6cr^4a+24a^2c^2-12ab^2r^2+12a^2cr^2+30br^4a-12ac^2r^2-48a^2cb)p^2
      +(-16ac^3-48acb^2-48ab^2r^2+48ac^2b+12br^4a+48cbar^2+16ab^3+12cr^4a)p
      -16c^3b-5br^6+3cr^6-16cb^3-20b^3r^2+24c^2b^2-4c^3r^2+36cb^2r^2+18cr^4b-3c^2r^4+21b^2r^4+4c^4-12c^2br^2+4b^4

    w = r(a^2r^12-48a^8+22a^3r^10+264a^5r^6-119a^4r^8-312a^6r^4+192a^7r^2)p^8
      +r(768a^4br^6-1344a^5br^4-8a^2r^10b+1344a^5cr^4+8a^2r^10c+184a^3r^8c-384a^7b-1152a^6cr^2-768a^4cr^6-184a^3r^8b+1152a^6br^2+384a^7c)p^7
      +r(-5760a^5cbr^2-432a^5cr^4+720a^3c^2r^6-2256a^4c^2r^4-1344c^2a^6-1344a^6b^2+16a^2c^2r^8+16a^2b^2r^8-32a^2cr^8b+4512a^4cbr^4-1440a^3cbr^6-432a^5br^4+720a^3b^2r^6-2256a^4b^2r^4+2880a^5c^2r^2+2880a^5b^2r^2+2688a^6cb+192a^6cr^2+192a^6br^2+12a^2r^10b+12a^2r^10c+288a^4cr^6-60a^3r^8c+288a^4br^6-60a^3r^8b)p^6
      +r(576a^2bc^2r^6-5472a^3bc^2r^4+11520a^4bc^2r^2+96a^5cr^4-864a^3c^2r^6+1728a^4c^2r^4+1824a^3c^3r^4+2688a^5c^3-2688a^5b^3-3840a^4c^3r^2+3840a^4b^3r^2+336a^2c^2r^8-192a^2c^3r^6-336a^2b^2r^8+192a^2b^3r^6-576a^2cb^2r^6+5472a^3cb^2r^4-11520a^4cb^2r^2-96a^5br^4+864a^3b^2r^6-1728a^4b^2r^4-1824a^3b^3r^4-8064a^5c^2b+8064a^5cb^2-1152a^5c^2r^2+1152a^5b^2r^2+88a^2r^10b-88a^2r^10c-288a^4cr^6+280a^3r^8c+288a^4br^6-280a^3r^8b)p^5
      +r(-20ar^10cb+17280a^3c^2b^2r^2-11520a^3cb^3r^2-48acb^2r^8+96acb^3r^6-864a^2bc^2r^6+2592a^3bc^2r^4+2784a^2bc^3r^4-2880a^4bc^2r^2-11520a^3bc^3r^2-48abc^2r^8+96abc^3r^6-144c^2b^2r^6a-4176c^2b^2r^4a^2+48ac^3r^8-24ac^4r^6+48ab^3r^8-24ab^4r^6+864a^3c^2r^6-696a^4c^2r^4-2592a^3c^3r^4-3360a^4c^4-26r^10ac^2-26r^10ab^2+2ar^12b+2ar^12c-3360a^4b^4+13440a^4cb^3+2880a^4c^3r^2+2880a^3c^4r^2+2880a^4b^3r^2+2880a^3b^4r^2-74a^2c^2r^8+864a^2c^3r^6-696a^2c^4r^4-74a^2b^2r^8+864a^2b^3r^6-696a^2b^4r^4+1228a^2cr^8b+528a^4cbr^4-864a^2cb^2r^6+2592a^3cb^2r^4+2784a^2cb^3r^4-1728a^3cbr^6-2880a^4cb^2r^2+864a^3b^2r^6-696a^4b^2r^4-2592a^3b^3r^4+13440a^4c^3b-20160a^4c^2b^2-26a^2r^10b-26a^2r^10c-24a^4cr^6+48a^3r^8c-24a^4br^6+48a^3r^8b)p^4
      +r(960ac^3r^4b^2-7680a^3cb^3r^2-5760a^2cb^4r^2-24acb^2r^8-576acb^3r^6+480acb^4r^4+2592a^2bc^2r^6-2016a^3bc^2r^4-3456a^2bc^3r^4+7680a^3bc^3r^2+5760a^2bc^4r^2+24abc^2r^8+576abc^3r^6-480abc^4r^4-960c^2b^3r^4a-1152a^2c^5r^2+280ac^3r^8-288ac^4r^6+96ac^5r^4+1152a^2b^5r^2-280ab^3r^8+288ab^4r^6-96ab^5r^4-192a^3c^2r^6+1824a^3c^3r^4-88r^10ac^2+88r^10ab^2+2688a^3c^5-2688a^3b^5+13440a^3cb^4-3840a^3c^4r^2+3840a^3b^4r^2+336a^2c^2r^8-864a^2c^3r^6+1728a^2c^4r^4-336a^2b^2r^8+864a^2b^3r^6-1728a^2b^4r^4+11520c^2b^3r^2a^2-11520a^2c^3b^2r^2-2592a^2cb^2r^6+2016a^3cb^2r^4+3456a^2cb^3r^4+192a^3b^2r^6-1824a^3b^3r^4+26880a^3c^3b^2-13440a^3c^4b-26880a^3c^2b^3)p^3+r(-168ar^10cb-1152acb^5r^2-864ac^3r^4b^2-1152ac^5br^2-8640a^2cb^4r^2+1356acb^2r^8-1152acb^3r^6+1296acb^4r^4-720a^2bc^2r^6+3840a^2bc^3r^4-8640a^2bc^4r^2+1356abc^2r^8-1152abc^3r^6+1296abc^4r^4+1728c^2b^2r^6a-3168c^2b^2r^4a^2-864c^2b^3r^4a+2880a^2c^5r^2-60ac^3r^8+288ac^4r^6-432ac^5r^4+2880a^2b^5r^2-60ab^3r^8+288ab^4r^6-432ab^5r^4+192c^6r^2a-1344a^2b^6+12r^10ac^2+12r^10ab^2+2880ac^4b^2r^2-3840c^3b^3ar^2-1344a^2c^6+16a^2c^2r^8+720a^2c^3r^6-2256a^2c^4r^4+16a^2b^2r^8+720a^2b^3r^6-2256a^2b^4r^4-20160a^2c^4b^2+26880a^2c^3b^3+8064a^2c^5b-20160a^2c^2b^4+8064a^2cb^5+192ab^6r^2+5760c^2b^3r^2a^2+2880c^2b^4r^2a+5760a^2c^3b^2r^2+160a^2cr^8b-720a^2cb^2r^6+3840a^2cb^3r^4)p^2
      +r(-4608acb^5r^2+3072ac^3r^4b^2+4608ac^5br^2+72acb^2r^8-1536acb^3r^6+3264acb^4r^4-72abc^2r^8+1536abc^3r^6-3264abc^4r^4-3072c^2b^3r^4a+184ac^3r^8-768ac^4r^6+1344ac^5r^4+8064ac^5b^2-13440ac^4b^3-2688ac^6b+13440ac^3b^4-8064ac^2b^5-184ab^3r^8+768ab^4r^6-1344ab^5r^4+2688ab^6c-1152c^6r^2a+384ac^7-384ab^7+8r^10ac^2-8r^10ab^2-5760ac^4b^2r^2+1152ab^6r^2+5760c^2b^4r^2a)p
      +r(b^2r^12+c^2r^12+2br^12c+630c^2b^2r^8+264c^5r^6-312c^6r^4-1344c^6b^2+2688c^5b^3+384c^7b-3360c^4b^4+2688c^3b^5+192c^7r^2-1344c^2b^6-48b^8-119b^4r^8+264b^5r^6-312b^6r^4+384b^7c+22c^3r^10+22b^3r^10+192b^7r^2-119c^4r^8+1728c^2b^5r^2+452c^3r^8b+1008c^5r^4b-1224c^4r^4b^2+528c^3r^6b^2+1056c^3r^4b^3-792c^4br^6-960c^6br^2-960c^4b^3r^2+1728c^5b^2r^2-960c^3b^4r^2+452cb^3r^8-792cb^4r^6+1008cb^5r^4-94cr^10b^2-94br^10c^2-960b^6r^2c-48c^8-1224c^2b^4r^4+528c^2b^3r^6)
    
    condition: a+b+c=r^2

 
Proof.

ax^4 + by^4 + cz^4 = w^2....................................................(1)

Let x = pt+1, y = t+1, z = t-1, w = gt^2+ht+r...............................(2)

Substitute (2) to (1) and using a+b+c=r^2, and simplifying (1), we obtain

(b+ap^4+c-g^2)t^4+(-4c+4b-2hg+4ap^3)t^3
+(-2rg-h^2+6c+6b+6ap^2)t^2
+(-2rh+4ap-4c+4b)t..........................................................(3)

Equating to zero the coefficient of t and t^2, then we obtain

g = (-2a^2p^2+4apc-4apb-2c^2+4cb-2b^2+3cr^2+3br^2+3ap^2r^2)/(r^3)

h = 2(ap-c+b)/r

Finally, we obtain t as follows

t = -4((2a^3-3a^2r^2+ar^4)p^3+(-6a^2c-3abr^2+6a^2b+3acr^2)p^2+(-12acb+6ab^2-3abr^2+6ac^2-3acr^2)p-cr^4+br^4-3b^2r^2-2c^3+2b^3-6cb^2+3c^2r^2+6c^2b)r^2
/((-4a^4+12a^3r^2+ar^6-9a^2r^4)p^4+(24a^2br^2-24a^2cr^2+16a^3c-16a^3b)p^3+(-24a^2c^2+48a^2cb-18br^4a+12ab^2r^2-24a^2b^2+12ac^2r^2+12a^2cr^2-18cr^4a-24cbar^2+12a^2br^2)p^2
+(48acb^2+16ac^3-16ab^3-48ac^2b-24ac^2r^2+24ab^2r^2)p-12c^2br^2-4c^4+16cb^3+12b^3r^2-24c^2b^2-18cr^4b+16c^3b+cr^6-12cb^2r^2+12c^3r^2+br^6-4b^4-9c^2r^4-9b^2r^4)

Substitute g, h, and t to (2), and obtain a parametric solution.            


   
Q.E.D.　
 
　　                  
       
3.Example


Case1: (a,b,c)=(1,1,2), x^4 + y^4 + 2z^4  = w^2

x = -33*p^4-44*p^3-54*p^2-140*p-241

y = -9*p^4-44*p^3-198*p^2+52*p-313

z = 9*p^4-4*p^3+150*p^2+188*p+169

w = 2(549*p^8+1464*p^7+3244*p^6+8136*p^5+28862*p^4+33160*p^3+108268*p^2+18040*p+60421)

Case2: (a,b,c)=(1,3,5), x^4 + 3y^4 + 5z^4  = w^2

x = -239*p^4-239*p^3-444*p^2-1559*p-4679

y = 13*p^4-302*p^3-1497*p^2+448*p-5822

z = -13*p^4-202*p^3+1119*p^2+1280*p+3536

w = 9(6347*p^8+12694*p^7+30311*p^6+103918*p^5+301040*p^4+690118*p^3+4652471*p^2+574654*p+7623647)





 




4.Reference

[1].Ajai Choudhry: Diophantine Equationf ax^5+by^5+cz^5=au^5+bv^5+cw^5, Rocky Mountain Journal of Mathematics. 29(1999)