(m^2+1)^2(m^2-1)X^4 + (m^2-1)Y^4 + (m^2+1)Z^4 = 2m^2W^4 Existence of solution for diophantine equation ax^4 + by^4 + cz^4 + dw^4 = 0 are known if abcd is square number. So, we are curious about whether above equation has a solution or not if abcd is not square number. In particular, when does this equation have infinitely many integer solutions? Parametric solutions of x^4 + ay^4 = z^4 + bt^4 are given below. x^4 + ay^4 = z^4 + bt^4 We show diophantine equation (m^2+1)^2(m^2-1)X^4 + (m^2-1)Y^4 + (m^2+1)Z^4 = 2m^2W^4 has infinitely many integer solutions. m is arbitrary. (m^2+1)^2(m^2-1)X^4 + (m^2-1)Y^4 + (m^2+1)Z^4 = 2m^2W^4.......................................(1) We use an identity p(t+1)^4+q(t)^4+r(t^2+at+b)^2=s(ct^2+dt+e)^2,..............................(2) with (a,b,e,p,q,r)=(2, d/(-c+d), d, -1/2sd^2(d-2c)/c, -1/2sc(d-2c), 1/2sd(d^2+c^2-2cd)/c). So, we look for the integer solutions {Z^2 = t^2+2t+d/(-c+d), W^2 = ct^2+dt+d}................(3) By parameterizing the second equation and substituting the result to first equation, then we obtain quartic equation below. Let (c,d)=(m^2,m^2+1), then u^2 = m^2k^4+(-2m^4+4m^2)k^2+(4m^3-4m)k+m^6+m^4-2m^2+1........................................(4) This quartic equation is birationally equivalent to an elliptic curve below. Y^2+(8m^4-8m^2)Y = X^3+(-2m^4+4m^2)X^2+(-4m^8-4m^6+8m^4-4m^2)X+8m^12-8m^10-32m^8+40m^6-16m^4 The corresponding point is P(X,Y)=( 2m^4-4m^2, -8m^4+8m^2 ). Hence we get 2P(X,Y)=( 17/4m^4+3/2m^2+1/4, -45/8m^6-101/8m^4+17/8m^2+1/8 ). This point P is of infinite order, and the multiples nP, n = 2, 3, ...give infinitely many points. This quartic equation has infinitely many parametric solutions below. n=2: X = 8(3m^2+1)m Y = 15m^4+18m^2-1 Z = 9m^4+22m^2+1 W = 21m^4+14m^2-3 n=3: X = 8(3m^2+1)(-3+14m^2+21m^4)m(1+22m^2+9m^4) Y = (15m^4+18m^2-1)(351m^8+420m^6+154m^4+100m^2-1) Z = 729m^12+11106m^10+16383m^8+4924m^6-569m^4+194m^2+1 W = 4941m^12+13338m^10+13467m^8+3212m^6-1997m^4-198m^2+5 . . etc. Hence we can obtain infinitely many integer solutions for equation (1). Example: m=2: 75X^4+3Y^4+5Z^4=8W^4: see details 3X^4 + 75Y^4 + 5Z^4 = 8W^4 m=3: 25X^4+4Y^4+5Z^4=9W^4

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