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New forms of matter, such as dark matter, seek new mathematical approaches. Here we describe an approach through the bicubic equation of limiting particle velocity formalism. The bicubic equation discriminant D in this undertaking satisfy 0 D determined by the congruent parameter z satisfying 2 1 z , where formally ( ) 2 3 3 2 z m m E  = , with m,  , and E being respectively, particle mass, velocity and energy. Also, nonlinearly related to the particle congruent parameter z is the particle congruent angle α. These two dimensionless parameters z and α simplify expressions in the bicubic equation limiting particle velocity formalism when evaluating, in both quadratic and linear forms, the three particle limiting velocities, 1 c , 2 c and 3 c ; (primary, secondary and tertiary) in terms of the ordinary particle velocity,  , in either quadratic or linear forms. Corresponding to these limiting velocities, for fixed values of congruent parameter ( ) z m and congruent angle ( ) m  , one then deduces, with equal values, dark matter particle energies E ( ) 1 c , E ( ) 2 c and E ( ) 3 c naturally in terms of limiting velocity squares 2 1 c , 2 2 c and 2 3 c . The exemplary values of the congruent parameter are in these regions, 1 3.53 z and / 2 / 3.25    . Already within these ranges of congruent parameters, the bicubic formalism yields for particle limiting velocities that 1 c and 2 c are complex conjugate to each other, * 1 2 c c = ,and that 3 c is imaginary. Furthermore, as long as z and α are kept fixed, the imaginary portions of 2 1 c and 2 2 c do not change the realities of numerically equal to each other dark matter energies ( ) i E c , i = 1; 2; 3: In fact, real ( ) 1,2 E c energies can be equally evaluated with 2 1,2 c or 2 1,2 Rec or even with 2 1,2 Imc so that in new notation, ( ) ( ) ( ) ( ) 2 2 2 2 1,2 1,2 1,2 3 Re Im E c E c E c E c = = = all with the same real values, with fixed values of ( ) z m and related ( ) m  . However, any individual of these energies’ changes if just in it the simultaneous changes in ( ) z m and ( ) m  are carried out, exhibiting the role of the particular limiting velocity square.

Real Energy Dark Matter Particles with Mostly complex Limiting Velocities in Either Quadratic or Linear Forms