Let a d = b c, then Hirschhorn's 3-7-5 identity, inspired by the Ramanujan 6-10-8 identity, is given by 25[(b + c + d)^3 + (a - d)^3 - (a + b + c)^3 - (c + d + a)^3 - (b - c)^3 + (d + a + b)^3][(b + c + d)^7 + (a - d)^7 - (a + b + c)^7 - (c + d + a)^7 - (b - c)^7 + (d + a + b)^7] = 21[(b + c + d)^5 + (a - d)^5 - (a + b + c)^5 - (c + d + a)^5 - (b - c)^5 + (d + a + b)^5]^2. Another version of this identity can be given using linear forms.

Eisenstein integer | Ramanujan 6-10-8 identity